There is no need for the solution $\psi(x)$ to be real. What must be real is the probability density that is "carried" by $\psi(x)$. In some loose and imprecise intuitive way, you may think about a TV image carried by electromagnetic waves. The signal that travels is not itself the image, but it carries it, and you can recover the image by decoding the signal properly.
Somewhat similarly, the complex wave function that is found by solving Schrödinger equation carries the information of "where the particle is likely to be", but in an indirect manner. The information on the probability density $P(x)$ of finding the particle is recovered from $\psi(x)$ simply by multiplying it times its complex conjugate:
$$\psi(x)^*\psi(x) = P(x)$$
that gives a real function as a result. Note that it is a density: what you compute eventually is the probability of finding the particle between $x=a$ and $x=b$ as $\int_{a}^{b} P(x) dx$
As you know, when you multiply a complex number(/function) times its complex conjugate, the information on the phase is lost:
$$\rho e^{i \theta}\rho e^{-i \theta}=\rho^{2}$$
For that reason, in some places one can (not quite correctly) read that the phase has no physical meaning (see footnote), and then you may wonder "if I eventually get real numbers, why did not they invent a theory that directly handles real functions?".
The answer is that, among other reasons, complex wave functions make life interesting because, since the Schrödinger equation is linear, the superposition principle holds for its solutions. Wave functions add, and it is in that addition where the relative phases play the most important role.
The archetypical case happens in the double slit experiment. If $\psi_{1}$ and $\psi_{2}$ are the wave functions that represent the particle coming from the hole number $1$ and $2$ respectively, the final wave function is
$$\psi_{1}+\psi_{2}$$
and thus the probability density of finding the particle after it has crossed the screen with two holes is found from
$$P_{1+2}= (\psi_{1}+\psi_{2})^{*}(\psi_{1}+\psi_{2}) $$
That is, you shall first add the wave functions representing the individual holes to have the combined complex wave function, and then compute the probability density. In that addition, the phase informations carried by $\psi_{1}$ and $\psi_{2}$ play the most important role, since they give rise to interference patterns.
Comment: Feynman is quoted to have said "One of the miseries of life is that everybody names things a little bit wrong, and so it makes everything a little harder to understand in the world than it would be if it were named differently." It is quite similar here. Every book says that the phase of the wave function has no physical meaning. That is not 100% correct, as you see.
Best Answer
The solutions are like sines and cosines (oscillating) when the energy of the particle is greater than the energy of the potential. Those regions are regions where a classical particle can exist. The solutions are like exponentials when the the energy of the particle is lower than the potential, regions where a classical particle cannot exist.
For example, if a particle with energy $E$ is in a region of no potential $V(x)=0$, the wave function will be sine-like. Suppose that particle encounters a barrier of height $V>E$. A classical particle does not have enough energy to pass by that barrier. The region of higher potential is classically forbidden. The particle will bounce off the barrier. But in quantum mechanics, the wave function extends into the barrier $\ldots$ into the forbidden region. In the forbidden region the wave function will fall exponentially.