Learn Russian: Мой брат избегает высоты, потому что он боится. - My brother avoids heights because he is afraid.

The verb избегать (to avoid) falls into the category of verbs dealing with fear, avoidance, or desire. In Russian, these verbs require their object to be in the Genitive case rather than the standard Accusative. Therefore, the nominative высота changes to the Genitive высоты.

The English translation says 'heights', so why is высоты in the singular form?

Высоты is the Genitive singular of высота (height). While English speakers say 'afraid of heights' in the plural, Russians usually use the singular to refer to the general concept of height. The plural Genitive would be высот, but it is less commonly used for this specific phobia.

The lesson says verbs of fear like бояться take the Genitive case. Why isn't there a Genitive word after боится?

You only use the Genitive case when you actually state the object the person is afraid of. In this sentence, the object is left out because it is already obvious from the first half of the sentence. Just like in English, you can simply say 'he is afraid' without repeating 'of heights'.

As a quick refresher, why is боится spelled without a soft sign (ь) before the -ся ending?

The soft sign is only used in the dictionary infinitive form, бояться (to be afraid). Because боится is conjugated for 'he' in the present tense, the soft sign is dropped. A common trick: if the question word for the verb is 'что делает?' (what is he doing?), there is no soft sign. If it is 'что делать?' (what to do?), it keeps the soft sign.