If Mandarin has a lot of homophones, how are the different meanings understood while speaking?

There’s no shortage of Chinese speakers here, and they’ll give better informed answers than me. But:

Mandarin Chinese is not Classical Chinese. Classical Chinese was a bit of a scholarly game, and writers relished the ambiguity of the homophones and the overall oracularity of it all. People in real life don’t, and Mandarin has dealt with homophony the way many languages do, by adding disambiguating words. Though people still have fun with Homophonic puns in Mandarin Chinese.

So the word for bat, fú 蝠, is homophonous with the word for good fortune, fú 福, and as a result bats commonly feature in Chinese art. But people who actually speak the language don’t call bats fú. They call them 蝙蝠 biānfú, combining two words for bat.

For another instance of ambiguity, look at Megan Cox’s answer to What are some homophones in Mandarin Chinese?. As Megan points out, there is homophony between bīng 冰 ‘ice’ and bìng 病 ‘illness, esp. mental illness’.

That’s not as homophonous as it gets; bīng 兵 (soldier) is a true homophone, and Wikipedia’s article on homophonic puns reports that in 1882, when there was fear of rebellions around Beijing, the sale of ice was banned as a result.

But even with that near homophony of bīng and bìng, Megan as a learner of Chinese may have been confused, yelling 你有病吗？ “Have you got a mental illness?” at the convenience store when she thought she was asking for ice. But the shop owner worked out what was going on, and he wouldn’t have been confused if she was fluent in Chinese. Ice as a noun is not bīng 冰 , but bīngkuài 冰块 ‘ice piece, ice cube’. So it would never be ambiguous with the noun bìng 病 ‘illness’.