Grammar induction is like teaching a machine how to talk like a human. Imagine you have a toy robot that can only say a few words like "hello" and "goodbye". You want to teach it to talk like you do, but you don't want to tell it every possible sentence it could say. Grammar induction helps the robot figure out how to make sentences on its own.
It's like giving the robot a set of rules to follow. These rules are called a grammar. A grammar tells the robot how to put words together in a sentence. It might say something like "a sentence can start with a noun or a verb". So the robot tries that out, and it might say "I see the cat" or "Run fast".
But there are lots of different possible rules for making sentences, and some sound more like what a human would say than others. So we want the robot to figure out which rules make the most sense. To do this, we give it lots of examples of normal human speech, and see which rules match up with those examples.
The robot might start to notice that sentences usually have a subject (who or what is doing the action), a verb (what the action is), and an object (what the action is being done to). It might also notice that adjectives (describing words like "blue" or "big") often come before nouns. Once it figures out these basic rules, it can start trying to make more complex sentences.
So with grammar induction, we guide the robot to learn the rules for making sentences by giving it examples to learn from. And once it knows the rules, it can make its own sentences that sound more like what a human would say.