I'll give a more grasp the concept answer if you don't mind. Consider the discovery of the planet Neptune.
Neptune was discovered because it was observed that Uranus wasn't moving as fast as orbital mechanics suggested it should be moving. By calculating how much Uranus had slowed down, an estimate was made for where an undiscovered planet should be and Neptune was identified very soon after.
Uranus is well inside Neptune's orbit, nearly twice as close to the sun and very far inside of Neptune's L1, but Neptune was still able affect Uranus' orbit enough to get noticed. Props to Newtonian mechanics. It works exceptionally well for a lot of measurements.
Loosely speaking, this type of orbital variation, L1 semi-stability or orbital perturbations like the slowing of Uranus' movement is a subset of the 3-body problem and that's actually more real. Observed orbits don't precisely follow Newton's or Kepler's formulas because of other massive bodies and perturbations. 3-body problem math gets crazy complicated, but the specific example of L1 can be explained using basic arguments without much math.
Earth's L1 point is about 1% closer to the sun or about .99 Astronomical Units or "AU". 1% closer gives an orbital velocity of a little over 0.5% faster and an orbital period of a bit over 1.5% faster or an orbital period of about 359.8 days, about 5.4 days shorter. That's what .99 AU circular Kepler orbit should do.
Consider an object at .99 AU, and a period of 359.8 days orbiting the Sun just inside the Earth's orbit - assuming both with circular orbits. Now imagine that object passes the Earth. It's relatively small so it doesn't change the Earth's orbit much, but passing that close to the Earth would give it a considerable gravitational tug, basically tossing it out of it's orbit. .99 AU is too close for a 2nd Kepler orbit and as a result, it's quickly tossed into a new orbit.
Two bodies can't orbit a massive central body with just 1% variation in their distance unless both bodies are very low mass (for example, Janus and Epimetheus are a rare exception to this).
Earth doesn't have low mass, so an object with a Kepler orbit at .99 AU would get tossed about after just one pass. In other words, if the object orbited as fast as your "gut" tells you it should, then it wouldn't be stable at all.
Now consider an object at .99 AU but orbiting the sun too slowly to maintain it's orbit. Slower velocity means the object falls in towards the sun, speeding up as it falls, entering a new, shorter period and more elliptical orbit. That's where the balance comes in. The slower velocity means the object wants to fall towards the Sun but being between the Earth and the Sun, Earth's gravity pulls it towards the Earth. The point where those two forces are equal is the L1 point, where it's no longer a Kepler orbit, but a 3 body orbit, slower than a Kepler orbit, but pulled in opposite cancelling ways by the Earth and the Sun. The "too slow" orbit is still unstable, but it's considerably more stable than a Kepler orbit in the same location would be.
With Earth's perturbed elliptical orbit, L1 isn't really a point so much as a smallish region, but on a computer, if we give the Earth a perfect circular orbit around the Sun with no other planets, L1 becomes a point of perfect balance. For actual orbits, L1 is always unstable.
Despite it's instability, Lagrange points are very useful low maintenance areas to station telescopes because much less energy is needed to keep the telescopes there. This only works if the central body (the Sun) is at least 24.96 times the mass of the orbiting, less massive body (the Earth)