To Swell
To Get Excited
intransitive verb, godan verb
盛り上げる is "to heap up" - this has a similar idea, except now this is more like an excitement or swelling. As excitement heaps up in you, you are going to get excited. Note how 上がる is in here. This means something is happening to you (or something else), but you're not doing the action. When you get all sorts of excitement building up in you, you aren't really doing it, it's just happening on its own in reaction to something else.
The reading comes from 盛り and 上がる. Let the excitement build as you realize you know the reading already.
子どもっていうのは、どんなささいなことでも手当たり次第に盛り上がることができるのよね。
Kids can take the most mundane, random things and get so excited.