If the Einstein Field Equations are so hard to solve, how did Einstein know they were correct in the first place?

It's not uncommon that the equations to describe a system are fairly simple but finding solutions is very hard. The Navier-Stokes equations are a good example - there's a million dollars waiting for the first person to make progress in finding solutions.

In the case of relativity, it became clear to Einstein fairly quickly that a metric theory was required so the equation needed was one that gave the metric as a solution. Einstein tried several variations before settling on the GR field equation. I believe one of the factors that influenced him was when Hilbert pointed out that the GR field equations followed from an obvious choice for the gravitational action.

I'm not sure if Einstein himself ever found an analytic solution to his own equations. However he used a linearised form of the equation to calculate the precession of Mercury and to calculate the deflection of light. The precession of Mercury was already known by then, so he knew (linearised) GR gave the correct answer there, but he had to wait a few years for Eddington's measurement of the deflection of light (though to modern eyes it seems likely that Eddington got the answer he wanted!).

The first analytic solution was Schwarzschild's solution for a spherically symmetric mass.

General relativity is one of the very few cases in science where a successful theory was devised purely on intellectual grounds rather than as a response to experimental data. Anyone who has suffered the pain of trying to learn GR can appreciate what an astonishing accomplishment this was, and why Einstein deserves every bit of the fame associated with him.

You can read about "History of general relativity" and in particular its field eqs here. In case you want a more official source, you can read MTW's Gravitation (the bible of GR) which quotes parts of official papers by Einstein. Both accounts coincide.

In short Einstein guessed the equations based on physical and mathematical intuitions. This is often the case for new physical theories. His first guess published on Nov 18 1915 was slightly wrong: he thought the Ricci tensor (a measure of curvature) was proportional to the stress energy tensor (a measure of the mass energy content). It also required some extra (bold, in his words) hypotheses. He managed to calculate the precession of Mercury perihelion and it was (surprisingly) correct. A few days later , however on Dec 2 1915 he published the eqs in its definitive form, getting rid of the extra, bold, hypotheses, and following very compelling logical/mathematical/physical/aesthetic steps. The calculation of the precession of Mercury perihelion did not change and so was still consistent with observations. This time he wrote that the Ricci tensor was proportional to a combination of the stress energy tensor and its trace. We nowadays write the eqs in its mathematically equivalent way as: the Einstein tensor (a combination of the Ricci tensor and its trace) is proportional to the stress energy tensor.

A few days earlier Hilbert (session of Nov 20 1915) managed to derive the field eqs using a simple lagrangian and the principle of least action. Apparently this was not known to Einstein.

Einstein was not able to solve the eqs for a star (a simple spherical source of gravity), but this was first solved by Schwarzschild in 1916.

Einstein could not find exact solutions to his would be eqs. However he used to linearize them, so he could find approximate solutions to weak gravitational fields, the same fields that we find in our solar system. This is how he managed to correctly calculate the precession of the orbit of Mercury