Learn Russian: Я кладу книгу.

In Russian, the object receiving the action (the direct object) takes the Accusative case. For feminine nouns ending in -а like книга, the ending changes to -у to become книгу.

The lesson is about inanimate objects. Why does the ending of книгу change if it is not alive?

The rule that inanimate objects don't change in the Accusative case only applies to masculine and neuter nouns! Feminine nouns ending in -а or -я always change their ending in the Accusative case, whether they are alive or just an inanimate object like a book.

Is кладу the only way to say 'I put' in Russian?

Russian is very specific about how things are placed! The verb класть (where кладу comes from) specifically means to put something down horizontally or lay it flat. If you were standing the book upright on a shelf, you would use a completely different verb (ставить).