How can astronomers say that the universe is expanding without knowing where the boundaries of the universe are?
There is something of a boundary to the universe, the Cosmic Microwave Background. It represents a point in the universe's history at which the universe was opaque, therefore we can't see anything beyond it. But we can't exactly measure how much it recedes every year, since it's a few billion light-years away, a distance that's not going to change by a measurable amount over the course of our lives.
Still, you haven't found a fatal flaw in cosmology, because there are always other tools at a scientist's disposal. I invoke the power of Doppler shift (or redshift, if you prefer). While we can't necessarily measure giant distances with high precision, we can usually get a reasonably precise measurement of the spectrum of an object. You may be used to thinking of a spectrum in terms of prisms and rainbows and such, and it's not far off. However, the spectra astronomers use tend to be much more detailed, to the point where we can make out individual spectral lines.
Without going into overly boring detail, a spectral line is a particular wavelength of light (a very specific shade of red, for instance) that is preferentially emitted by an individual atom. You can learn more: here, along with a bunch of other quantum physics stuff, if you're interested.
Back to Doppler shift, when an object is in motion, the spectral lines emitted by the atoms move around to different wavelengths. It's similar to the effect that causes a train's whistle or an ambulance's siren to change pitch as it goes flying past you. If an object is moving towards you, the light gets shifted toward shorter wavelengths; if it's moving away, the light is shifted toward longer wavelengths. These effects are called blueshift and redshift, respectively. (Though, to be sure, if light is already blue, it will get blueshifted toward violet or ultraviolet, and if light is already red, it will get redshifted toward infrared. It gets confusing if you put too much thought into it, so just think about it in terms of what effect it has on yellow light.)
Knowing this, Edwin Hubble (the guy the telescope is named after) went out and measured the Doppler shift of several nearby galaxies. If you assume the universe is not expanding, you'd expect to see a more or less even distribution of blueshifts and redshifts. Surprisingly enough, he found that almost all of the galaxies he measured were redshifted, meaning every galaxy is traveling away from us. Even more evidence has been amassed today, and we know that galaxies very far away are extremely redshifted. So we know beyond a shadow of a doubt that everything in the universe seems to be moving away from everything else. And if this is the case, the universe must be expanding, as it's hard to imagine a way to make things get farther apart without actually increasing the extent of space they cover.
To get mathematical for a moment, prior to Hubble's measurements, Albert Einstein had formulated his theory of General Relativity, coming up with an equation that described the entire universe. He was unsatisfied, however, when he realized that his equation predicted that the universe must be either expanding or contracting. He was able to create a solution in which the universe could be stable, but upon the announcement of Hubble's discovery, he changed his mind, calling the alteration the biggest mistake of his career.
So, one of the greatest minds of all time predicted a universe that was expanding, and one of the first guys to go looking found he was right, as has everyone who's gone looking since. And hopefully, that's all the answer you need.