The doctors are worried that the baby might be a mutant (the possibility of a miscarriage is Rather Too Much for comics the late 1960s), and they're not wrong. Young little Henry has gigantic hands and feet. Despite his athletic abilities, he becomes a bit of a bookworm until he is press-ganged onto the varsity football team, and leads the side on a winning streak, reversing their fortunes.
When some criminals try to hold up the box office, Hank stops them (on live television), and in doing so attracts the attention of two mysterious figures: Chico and El Conquistador. They kidnap him, and his parents. El C blackmails Hank into helping them steal an experimental solar generator.
Meanwhile, Xavier has sent Iceman to investigate a new reading on Cerebro (quietly retconning away part of the story in #15), and found evidence of the abduction. Bobby reports back, and Professor Xavier is unable to locate McCoy more precisely. He unveils his new prototype mutant-detecting machine Cerebro, and uses that instead. (It may not surprise you to learn there was an issue break somewhere in the middle of that! I can't blame First Class and the other later implants for not getting Cerebro's creation right if original material forgets that it has already been introduced in the very same story it is later presented as a new invention in!) He dispatches the full team of X-Men - Cyclops, Iceman and Angel (whose backstory a footnote promises will be explained later). They rescue him. Etc. Xavier fixes everything with a mindwipe. Whether Beast's parents remember anything at all is left unclear...
Beast's origin story would get revisited in X-Men Origins: Beast in 2008 (released to coincide with the Wolverine movie of the following year). This story, by Mike Carey, who has form for revisiting X-Men continuity in his run on X-Men: Legacy (particularly the immediately post-Messiah Complex stuff, which deals with more than its fair share of Xavierdickery), keeps the basic structure of these backups intact. The details are changed - Beast rescues himself, and the Conquistador, although kept, now becomes the subject of in-story mockery from heroes and minions alike. But the real difference is the making clear of Hank's emotional journey, from trying to keep his head down, to finding a place to excel - and his decision to leave his old life behind, something implicit in the original material but never clearly expressed.