Some of those sentences used to be translated with "jumps in into the river". Which is quite odd, but almost helpful (not really) in understanding what is going on here.
The effect behind it is twofold: first, the verbal prefix, if attached to the verb, "catches" the focus. Meaning, in this sentence the focus lies on the action itself, which gives sentences like these a rather neutral tone.
Second, verbal prefixes give the action a perfective meaning. The monkey will indeed end up in the river (or has ended up there if the sentence is talking about the past). May make a bit more an obvious sense once you go to past tense: "Keltem." - I was getting up. I was in the process of getting out of bed, but it's not certain if I succeeded or if the pillow suddenly looked much more comfortable. "Felkeltem." - I got up. I am out of bed and the day can begin.
Let's play a bit with it:
- A majom beugrik a folyóba. - The monkey jumps into the river. It arrives there safely.
- A majom ugrik a folyóba. - It is the monkey that's making its way into the river.
- A majom a folyóba ugrik. - Where is the monkey jumping? Into the river.
- A majom ugrik be a folyóba. - It is the monkey that's going to be in the river.
- A majom a folyóba ugrik be. - The monkey enters the river, and not any other body of water.
- Beugrik a majom a folyóba. - It's jumping in, not just falling off the tree.
Examples 1 and 3 are the most common way to state what's happening. The other sentences shift the emphasis to certain words, so that those sentences rather function as answers to specific questions. Example 2 would be an answer to "Mi ugrik a folyóba?" - "What is jumping into the river?"
A simple way to see what the sentence emphasises is switching the word in front of the verb stem with the respective question word. Like I just did.