Aksakal's answer is correct. By controlling for all variables in a regression, you "keep them constant" and you are able to identify the partial correlation between your regressor of interest. Let me give you an example to make this clearer.
First, let us create some correlated $X$s.
ex <- rnorm(1000)
x1 <- 5*ex + rnorm(1000)
x2 <- -3*ex + rnorm(1000)
x3 <- 4*ex + rnorm(1000)
Now, since all these variables are generated by some underlying variable $ex$, they are clearly correlated. You can check this using cor(x1,x2), for instance.
Now, let us generate the dependent variable with known parameters.
y <- 1*x1 + 2*x2 + 3*x3 + rnorm(1000)
Here we know that $\beta_1=1, \beta_2=2, \beta_3=3$. I have picked them arbitrarily. Let us now see if Aksakal's approach can uncover these parameters:
lm(y ~ x1+x2+x3)
If it works, the estimated parameters should be close to the ones we have picked. Here the result:
Call:
lm(formula = y ~ x1 + x2 + x3)
Coefficients:
(Intercept) x1 x2 x3
-0.01224 0.99805 1.99746 2.99670
As you can see, all parameters have been uncovered.
Having said that, there are many caveats involved here as well. Most importantly, you should not interpret these coefficients in a causal way. Depending on your actual situation, it might help if you explain a bit more what you are trying to estimate so that people can evaluate whether this method is appropriate (or whether answering your research question is feasible at all). For instance, why do you think your independent variables are correlated? Is it that $X_1$ might have an effect on $X_2$ and this has an effect on $y$? If this is the setup you have in mind, then depending on your field, you may want to look into mediator/moderator analysis or into quasi-experimental methods. Hence you see you might benefit from elaborating a bit more on your situation.
If you are reasonably sure that there ought to be relationships between each of the IVs and the DV then you should plot the DV against each of the IVs. For the continuous IV you can try a scatterplot and add a loess line. For the dichotomous IVs, one fairly simple option is to jitter the IV and then do a parallel boxplot with the DV. With so much data, you will probably need quite a bit of jitter and you might also try making the points open circles.
See what that reveals.
For the full regression model, one possible option with a lot of IVs is a regression tree - this may reveal interesting interactions. If it looks promising, then you might investigate things like random forests.
Best Answer
In a linear model, the t-statistic show us if there is a significant linear relation between the independent variable. If the t-statistic is close to zero, there is no linear relation or the linear relation is so weak that can not been proven.