Why is there a strong attractive force btwn H atoms at short distances?

Answer 1

Because each hydrogen atom has only one electron in its valence shell. The tendency is to fill the shell, and since 2 is the maximum, they tend to couple and give us H2. Even though they are sharing each others electron, it gives both an effective two in the shell.

Answer 2

This is exactly what the chemical bond consist of.

Simplifying greatly, the electrons (negatively charged) sit in between both the nuclei, which are positively charged, and are attracted to both of them at once. This is an example of a covalent bond, which is formed by the sharing of electrons between atoms.