Explain why metal atoms form positive ions.

There is more than one way to look at this question. The energy required to remove an electron from an atom can be measured experimentally using photoelectron spectroscopy (PES). Less energy is required to remove the first and second electron from metal atoms than from nonmetal atoms. Since electrons are negatively charged, removing one or more electrons from a neutral atom leaves a positive ion. Frequently in chemistry and physics, processes that do not require much energy occur readily, although there can be many qualifications to this general rule. In the case of the formation of positive ions, normally this only occurs when a nonmetal atom gains an electron, forming a negative ion. This gives off energy, which can drive the removal of an electron from a metal atom.
We can also relate the relative ease of removal of electrons from metal atoms to other things we know about the atoms. Most metal atoms have only one or two valence electrons—electrons which, if removed, would leave the atom with a filled shell at the next lower energy level. Thus, for most metals, not much charge separation is required to yield a configuration having a stable valence shell. We can also look at the force of attraction between the valence electrons of metals and the nucleus. Because core electrons are generally closer to the nucleus, the valence electrons are “shielded” from the nuclear charge by these inner electrons, experiencing an “effective nuclear charge” that is much less than the actual nuclear charge—and also much less than the effective nuclear charge experienced by valence electrons in nonmetals. As if that was not enough, the atomic radius of metal atoms is relatively large. Since the attraction of oppositely charged particles is inversely proportional to the square of the distance between them, at distances typical for metal atoms, valence electrons feel a relatively weak attraction to the nucleus.
For all these reasons, metal atoms easily lose valence electrons, forming positive ions.