Made is wrong for two reasons, one grammatical and one semantic. (1) The object of "make" takes a bare infinitive, not a to-infinitive, just like "let". We say "Make those kids settle down," not "Make those kids
to settle down." (2) "Make" suggests compulsion: that the nomads didn't settle down of their own accord, but because someone or something else made them do so. This is nicely illustrated by a standard statement of childish defiance:
"You must return Bobby's comic book."
"Oh yeah, who's gonna make me?" […make me return the comic book, that is]
"Making" someone do something suggests manipulating them somewhat like an inanimate object. When someone says "Bobby made me angry", they're saying that they have no choice about their anger: Bobby caused it. Sometimes you can use this denotation of compulsion loosely or metaphorically, as in "The devil made me do it," but merely noticing that plants sprout from seeds doesn't seem quite strong enough to merit even a loose usage of "made".
Forced would be grammatically correct, but the meaning is wrong. "Force" in this context means that all but one of the choices is catastrophic, so in effect you have "no choice" but that one. For example:
When the car in front of me slammed on its brakes, I was forced to swerve into the next lane to avoid a collision.
The expanding glaciers forced the nomads to leave northern Europe [lest they starve to death].
The quotation suggests nothing so drastic. It suggests that the nomads could easily have continued hunting and gathering. Farming was not their only choice.
Decided isn't completely wrong, but it's jarring because, used transitively like that, it suggests a decision made deliberately—decisively, even. For example:
The adventure decided me to never again leave the limits of my prescribed stamping grounds until I was ready to venture forth for good and all, as it would certainly result in a curtailment of my liberties, as well as the probable death of Woola, were we to be discovered.
I chose an Edgar Rice Burroughs novel to illustrate this sense of "decided" because today it's a somewhat old-fashioned usage, but old-fashioned is not wrong. What's wrong is the suggestion that noticing that plants grow from seeds was itself decisive: that that alone may have settled the matter of whether to settle down as farmers. Usually when X "decides" someone to do Y, X is felt to be either a sufficient or overriding criterion, or an occasion that made the decisive criteria clear (like the "adventure" above). Discovering that plants can be cultivated might suggest seeing if it's possible to cultivate more of them, but the idea of switching over to agriculture, living entirely on cultivated crops, raises further questions, like whether that's feasible, how much work is involved, and how reliable it is. More investigation is needed, hence "plants grow from seeds" doesn't decide the matter.
Convinced is similar to "decided", but not old-fashioned, and more compatible with a gradual change of mind. It's also more compatible with pressures other than explicit reasoning, like its synonym persuaded. Unlike "decided", "convinced" suggests opposition and resistance. People are convinced to vote for a candidate, convinced to try a diet, convinced that O.J. was guilty, convinced of a scientific theory, etc.—by stronger evidence or factors than support an opposing candidate, diet, verdict, theory, etc. When a new opinion challenges your current opinion in a sort of mental battle, and wins, you are "convinced". Often the old opinion was just to continue as you were. The test author does seem to be suggesting that prehistoric people gave up nomadic life because the discovery of germination gave them hope of a more stable food supply in farming—a relative advantage. But, as with "decided", nomads would likely need to experiment further before becoming convinced to settle down as farmers. Reasonable skepticism about an entirely new way of life would not be overcome by "Hey, look, plants germinate!"
So, of the four answers given, I'd say convinced is best—though not fully satisfactory. A much better choice would be led. "Led" suggests that the nomads "followed" the opportunities for growing plants from seeds and roots, one thing "led" to another, and 20,000 years later their descendants were building permanent structures and cultivating yearly crops. It doesn't suggest that merely noticing that plants grow from seeds was enough to convince, persuade, decide, or get them to abandon hunting and gathering. Noticing germination was one step on a long path toward agricultural life, mostly unforeseen at the time of that step—as suggested by both anthropologists and the context given in the question.