how does a monosaccharide become a disaccharide and a polysaccharide?

By the formation of glycosidic bonds between adjacent monosaccharides - a dehydration reaction  -one H2O is formed per bond formed.

"glycosidic bond is a type of covalent bond that joins a carbohydrate (sugar) molecule to another group, which may or may not be another carbohydrate"

Both of these processes are made possible in living cells by enzymes. We have enzymes that bring monosaccharides together and catalyze the formation of glycosidic bonds, producing disaccharides or polysaccharides.

We also have enzymes that break glycosidic bonds, such as amylase, which is found in saliva. Amylase is the reason that if you chew a cracker for a while, it starts to taste sweet. The starch in the cracker, a polysaccharide, is being broken down by amylase into mono- and disaccharides, which our taste buds register as sweet.

You seem to have already gathered that the breakdown of sugars is a hydrolysis reaction. This means that the elements of water (H and OH) are added, one to each of the sugars that are being broken down. The sugars bond to these groups instead of to each other.

The reverse happens in the condensation reactions that form glycosidic bonds. In these reactions, the elements of water (again, H and OH) are removed, leaving behind a free O on one sugar and a free C on the other, which then bind to each other, forming a glycosidic bond.

Both of these types of reactions require an enzyme to happen at a useful pace. If you leave a bowl of sugar or starch sitting around, it is not going to react this way. Enzymes are proteins that are specially shaped and with special properties to speed up these reactions. They bring the sugar molecules together in just the right configuration, and then use  their own amino acids to transfer protons and OH groups so that the sugars will bind to each other. Or, to OH and H groups from the surrounding water (in the case of hydrolysis).

Hope this helps!