Why does gravity increase in star formation?

Some good answers, I'm going to give kind of summary, cause you touched on a few points.

Why does gravity increase in star formation

Gravitation is a product of a few forces. Mass, density and, not to be ignored, rotation speed.

It's not actually the fusion process that keeps the sun from contracting, at least, not directly. It's heat that keeps the star expanded. That's the balancing act. High temperature wants to expand, gravity wants to contract.

The fusion process is actually pretty slow, which is why stars like our sun have a main sequence of about 10 billion years, and a lot of the heat that a star starts out with is from the heat of formation. Potential energy gets converted to heat due to the coalescing and condensing of all that matter so stars start out hot, even before fusion begins.

In fact, a star in formation can be many times brighter than the star is during it's main sequence due to the high heat of formation. Here's an article that says the forming sun was 200 times brighter than it is now.

Young proto-stars, as a result of conservation of angular momentum, tend to rotate very fast and that fast rotation can create a bulge and increases ejection of matter. The formation process is pretty chaotic compared to the main sequence stage. Lots of ejected matter, much bigger solar storms, lots of lheat from formation, etc.

Once the main sequence stage is underway and rotation is slowed down, then there's more of a balance between heat and gravity mentioned above. The fusion process continues to add heat to the core of star which the star, convects or conduct heat away from the core into the outer layers and then, radiates from it's surface, but during the main sequence, in general, the core of the star gradually heats up and in most cases, the energy added from fusion isn't nearly strong enough to blow apart the star, unless the star is enormously large like over 150 or 200 solar masses, then the star doesn't really work without blowing off a bunch of matter. See: here.

I get that the fusion of hydrogen atoms releases energy... fine...

How does gravity keep it together if the mass is lessening as a result
of fusion( mass being converted into energy from fusion) while gravity
is weakening( as mass lessens )?

As others have said, mass loss by solar wind is a bigger factor especially for young and smaller stars, but there's a few factors at play. The short answer to this question is that the mass loss, at least by fusion, is quite very compared to the total mass of the star. Another factor, as hydrogen becomes helium, the core of the star becomes denser and greater density tends to be smaller and that increases gravity, but there are competing factors. The inner core grows denser as it becomes more hydrogen rich and the fusion tends to expand outwards on the outside of the helium core, so a star like our sun gets a denser inner core over time, but the layers around the core can grow hotter and larger, even as they lose mass.

Wouldn't the radiation overpower the force of gravity and tear the
star apart?

As mentioned above, this happens if you have 150 or 200 solar masses. lower mass stars, the fusion isn't nearly powerful enough to blow the star apart. Stars and white dwarfs blow apart when they go supernova, but that's different than the main sequence fusion process.

Our sun will blow off some of it's matter when it has it's helium flash, so there are examples of what you're describing happening, but not during the main sequence for stars like our sun when material is expelled primarily by magnetic storms causing coronal mass ejections. Fusion is, generally speaking, more like a slow burn, than a big explosion when it's up against the enormous gravitational binding energy of a star.