@Magma gives a beautiful self-contained answer — but to put it into a bigger picture, it’s worth also mentioning the word compactness, the term for a lot of phenomena like this one, and which gives this as a consequence of various quite general theorems.
As a logician, the first way that springs to mind for me is the compactness theorem for first-order logic: if every subset of some first-order theory has a model, then so does the whole theory. So: write down a first-order theory describing a suitable tiling of the plane; and note that “arbitrarily large squares can be tiled” implies that every finite subset is consistent. (There are lots of ways to write such a theory, but one approach is purely propositional: for each $i,j \in \mathbb{Z}$, take atomic propositions $H_{i,j}$ and $V_{i,j}$ in the language, read as saying that the square $(i,j)$ is in the same tile as its horizontal neighbour $(i+1,j)$ or its vertical neighbour $(i,j+1)$ respectively; and for each $(i,j)$, give an axiom $\alpha_{i,j}$ expressing that some copy of the polyomino is formed by tiles connected to $(i,j)$, and that these are not connected to any of their other neighbours.)
But compactness can be also seen in other guises, e.g. topologically. Intuitively, @Magma’s argument can be seen as saying that the sequence of arbitrarily large finite tilings must have a convergent subsequence within some suitable compact topological space of tilings, which must converge to a tiling of the whole plane. I don’t immediately what space to use, that would make each of the above steps “easy” (either directly or from general theorems) — but I’m sure someone more familiar with writing in such terms can see how to write it this way!