Once you know how far away a planet is, you can use the orbital periods (P) of moons circling a planet and how far the moons are from the planet (d) to measure the planet's mass. You measure the angular separation between the moon and the planet and use basic trigonometry to convert the angular separation into distance between the planet and moon. That conversion, though, first requires that the distance to the planet and moon be known.
Isaac Newton used his laws of motion and gravity to generalize Kepler's third law of planet orbits to cover any case where one object orbits another. He found for any two objects orbiting each other, the sum of their masses, planet mass + moon mass = (4pie^2/G) × [(their distance apart)^3/(their orbital period around each other)^2]. Newton's form of Kepler's third law can, therefore, be used to find the combined mass of the planet and the moon from measurements of the moon's orbital period and its distance from the planet.
You can usually ignore the mass of the moon compared to the mass of the planet because the moon is so much smaller than the planet, so Kepler's third law gives you the planet's mass directly. The one noticeable exception is Pluto and its moon, Charon. Charon is massive enough compared to Pluto that its mass cannot be ignored. The two bodies orbit around a common point that is proportionally closer to the more massive Pluto. The common point, called the center of mass, is 7.3 times closer to Pluto, so Pluto is 7.3 times more massive than Charon. Before the discovery of Charon in 1978, estimates for Pluto's mass ranged from 10% the Earth's mass to much greater than the Earth's mass. After Charon's discovery, astronomers found that Pluto is only 0.216% the Earth's mass---less massive than the Earth's Moon! For planets without moons (Mercury and Venus), you can measure their gravitational pull on other nearby planets to derive an approximate mass or, for more accurate results, measure how quickly spacecraft are accelerated when they pass close to the planets.