Xjyuk

I was reading Paul Renteln "MANIFOLDS, TENSORS, AND FORMS An Introduction for Mathematicians and Physicists" p.145, where he defined the Betti number as $dim H_m(K)$, where $H_m(K)$ is the quotient space of cycles modulo boundary $Z_m(K)/B_m(K)$. He said it is true that the Betti number counts the number of m-dimensional holes on K, but I don't see why.

This is similar to what Wikipedia said at https://en.wikipedia.org/wiki/Simplicial_homology, the last sentence of the definition section. Can anyone provide me with an intuitive explanation of what is going on?

This sentence in the paragraph just before the line from wikipedia you quote

It follows that the homology group Hk(S) is nonzero exactly when there
are k-cycles on S which are not boundaries. In a sense, this means
that there are k-dimensional holes in the complex.

provides an answer to your question. A $k$-cycle is (intuitively speaking) a geometric structure which potentially surrounds a $k+1$-dimensional region. $1$-cycles are easiest to visualize - think loops. On the surface of a sphere every loop can be thought of as having an inside (two, in fact, since there's no topological distinction between inside and outside). Every $1$-cycle is the boundary of a part of the surface and the first Betti number is $0$. There are no holes. On the surface of a torus there are two kinds of loops that don't have interiors - those are $1$-cycles that aren't boundaries, so a torus has $2$ "holes" where there might be disks but aren't. Those are conceptual holes - there is no hole that looks as if was cut out by a cookie cutter.

In higher dimensions you can ask whether something that looks like the surface of an ordinary sphere - a $2$-cycle - actually surrounds something like the interior of a sphere. If not, that $2$-cycle contributes to the second Betti number.

I hope this serves as the intuition you need. The technical details are, of course, technical.