Every rocket spends most of its effort accelerating its own empty tanks. Throwing away a tank once it is empty means the rest of the flight no longer has to carry it, and that is what makes orbit reachable. Build the same 500 metric ton rocket as one stage or two and compare what it can do.
What to try. Start with two stages, then switch to one and watch the total collapse: the same 500 metric tons of hardware and fuel loses roughly three kilometres per second, because the single stage has to drag its whole empty structure all the way to burnout. Then push the structure slider up towards 0.25, the sort of number a steel water tank would give, and see even the two-stage rocket fall short. Press "best split" to find the division that gets the most out of a given rocket. The masses come out lopsided, around 89% in the first stage, but that is exactly what makes the two stages contribute the same Δv, which is where the maximum sits. The first stage has to be far bigger because it carries the second stage as part of its payload.
When equal Δv is the answer, and when it is not. Splitting a rocket so that each stage supplies the same Δv is optimal only when the stages are alike: same exhaust speed, same structure fraction, and no gravity loss. Give the upper stage a better engine and the optimum shifts towards letting it do more of the work; turn gravity losses on and it shifts the other way, because the first stage loses far more to gravity than the second. The readout says which case you are in, and the sim finds the true maximum numerically either way.
About the climb angles. Gravity steals g sin γ for every second the engine burns, where γ is how steeply the rocket is climbing. Set the second stage to 0° and its loss disappears: that is why a launcher pitches over early and spends its upper-stage burn flying almost sideways. Set it to 90° instead and the same rocket cannot reach orbit at all.
Model: total liftoff mass fixed at 500 metric tons (500,000 kg); masses on the diagram are labeled in metric tons, written t. Δv = ve ln(minitial/mfinal) for each stage in turn, each one carrying everything above it, with the empty stage dropped at separation. Constant mass flow, so the burn time is mpropellantve/(Ψminitialg) and the gravity loss is g sin γ times that. No drag, no thrust variation with altitude, no coast between stages. The orbit target is 9.4 km/s in the ideal case, which is the usual allowance for a real launch including its losses, and 8.4 km/s once gravity is modelled explicitly (7.8 km/s of orbital speed plus about 0.6 for drag and steering).