The method is "keep trying". Try reading the paper again, try reading a different paper, try stopping reading papers and just work it out yourself. Try talking to other people, try struggling on your own, then try talking to people again. Try a different problem, then try going back to the same problem. Try the same method again and again, then try different methods, then go back to the first method and try it one more time. Try just writing out some calculations even if you don't see them going anywhere, try a stupid example, try drawing some pictures. Finally try taking a break, and then also try not taking a break. Try everything once, then try it again!
But enough of that, here's some more practical advice from my own experience (I'm a postdoc by the way).
1) For a hard problem, work intensely on it for a few weeks or months, until you feel like you've completely hit a wall. During this period you want to be producing as many calculations, lemmas, pictures, examples, etc as possible. They don't all have to be super-relevant -- your goal is maximum output of material. Then when you finally feel like you're going insane, take a deep breath and move on to something else. After a break of a few weeks or months, which may be spent either on work of some other kind or even on vacation, come back to the problem and see if the fresh perspective helps. It may take multiple cycles like this for a really hard problem.
2) Look for a variation of your problem which is easier or which your methods can better handle. Even if the variation is not as strong of a result as what you are ultimately aiming for, it may still "count" and end up being a nicer paper than you realize at first. It's good for your morale to register progress in this way, and if you can get a publication it's good for your CV too. Here is where advisors and mentors can help a lot -- they can help you identify which variation is meaningful, doable, and "interesting".
3) Remember that almost nothing ever works. One of the worst feelings to me is when I find a mistake in something I thought I had figured out, and all of a sudden weeks or months of progress begin to unravel. This can be extremely depressing. Try not to take it too hard, and over time you can rebuild and recover. Try also to appreciate the progress you are making, even if it is much less than you wish it was.
Good luck!
Wikipedia has a list of 21st-century mathematicians while the book Mathematicians : an outer view of the inner world has the following list in portrait form with their autobiography:
Adebisi Agboola --
Michael Artin --
Michael Francis Atiyah --
Manjul Bhargava --
Bryan John Birch --
Joan S. Birman --
David Harold Blackwell --
Enrico Bombieri --
Richard Ewen Borcherds --
Andrew Browder --
Felix E. Browder --
William Browder --
Lennart Axel Edvard Carleson --
Henri Cartan --
Sun-Yung Alice Chang --
Alain Connes --
John Horton Conway --
Kevin David Corlette --
Ingrid Chantal Daubechies --
Pierre Deligne --
Persi Warren Diaconis --
Simon Donaldson --
Noam D. Elkies --
Gerd Faltings --
Charles Louis Fefferman --
Robert Fefferman --
Michale Freedman --
Israel Moiseevich Gelfand --
William Timothy Gowers --
Phillip Griffiths --
Mikhael Leonidovich Gromov --
Benedict H. Gross --
Robert Clifford Gunning --
Eriko Hironaka --
Heisuke Hironaka --
Friedrich E. Hirzebruch --
Vaughan Frederick Randal Jones --
Nicholas Micahel Katz --
Robion Kirby --
Frances Kirwan --
Joseph John Kohn --
János Kollár --
Bertram Kostant --
Harold William Kuhn --
Robert Phelan Langlands --
Peter David Lax --
Robert D. MacPherson --
Paul Malliavin --
Benoit Mandelbrot --
William Alfred Massey --
John N. Mather --
Barry Mazur --
Margaret Dusa McDuff --
Curtis McMullen --
John Willard Milnor --
Maryam Mirzakhani --
Cathleen Synge Morawetz --
David Mumford --
John Forbes Nash, Jr. --
Edward Nelson --
Louis Nirenberg --
George Olatokunbo Okikiolu --
Kate Adebola Okikiolu --
Andrei Okounkov --
Roger Penrose --
Arlie Petters --
Marina Ratner --
Kenneth Ribet --
Peter Clive Sarnak --
Marcus du Sautoy --
Jean-Pierre Serre --
James Harris Simons --
Yakov Grigorevich Sinai --
Isadore Manual Singer --
Yum-Tong Siu --
Stephen Smale --
Elias Menachem Stein --
Dennis Parnell Sullivan --
Terence Chi-Shen Tao --
Robert Endre Tarjan --
John T. Tate --
William Paul Thurston --
Gang Tian --
Burt Totaro --
Karen Keskulla Uhlenbeck --
Sathamangalam Rangaiyengar Srinivasa Varandhan --
Michèle Vergne --
Marie-France Vigneras --
Avi Wigderson --
Andrew John Wiles --
Shing-Tung Yau --
Don Zagier.
Best Answer
My advice would be:
$\bullet $ Do many calculations
$\bullet \bullet$ Ask yourself concrete questions whose answer is a number.
$\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!)
$\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in your subject.
I have answered many questions tagged algebraic geometry on this site and I was struck by the contrast between the excellent quality of the beginners in that field and the nature of their questions: they would know and really understand abstract results (like, say, the equivalence between the category of commutative rings and that of affine schemes) but would have difficulties answering more down-to-earth questions like: "how many lines cut four skew lines in three-dimensional projective space ?" or "give an example of a curve of genus $17$".
In summary the point of view of some quantum physicists toward the philosophy of their subject
Shut up and calculate ! contains more than a grain of truth for mathematicians too (although it could be formulated more gently...)
Nota Bene
The above exhortation is probably due to David Mermin, although it is generally misattributed to Richard Feynman.
Edit
Since @Mark Fantini asks for more advice in his comment below, here are some more (maybe too personal!) thoughts:
$\bigstar$ Learn mathematics pen in hand but after that go for a stroll and think about what you have just learned. This helps classifying new material in the brain, just as sleep is well known to do.
$\bigstar \bigstar$ Go to a tea-room with a mathematician friend and scribble mathematics for a few hours in a relaxed atmosphere.
I am very lucky to have had such a friend since he and I were beginners and we have been working together in public places ( also in our shared office, of course) ever since.
$\bigstar \bigstar \bigstar$ If you don't understand something, teach it!
I had wanted to learn scheme theory for quite a time but I backed down because I feared the subject.
One semester I agreed to teach it to graduate students and since I had burned my vessels I really had to learn the subject in detail and invent simple examples to see what was going on.
My students did not realize that I was only one or two courses ahead of them and my teaching was maybe better in that the material taught was as new and difficult for me as it was for them.
$\bigstar \bigstar \bigstar \bigstar$ Last not least: use this site!
Not everybody has a teaching position, but all of us can answer here.
I find using this site and MathOverflow the most efficient way of learning or reviewing mathematics . The problems posed are often quite ingenious, incredibly varied and the best source for questions necessitating explicit calculations (see points $\bullet$ and $\bullet \bullet$ above).
New Edit (December 9th)
Here are a few questions posted in the last 12 days which I find are in the spirit of what I recommend in my post: a), b), c), d), e), f), g), h).
Newer Edit(December 17th)
Here is a fantastic question, brilliantly illustrating how to aggressively tackle mathematics, asked a few hours ago by Clara: very concrete, low-tech and naïve but quite disconcerting.
This question also seems to me absolutely original : I challenge everybody to find it in any book or any on-line document !