Thanks for your question, I will be more than happy to help clarify any doubts you might have!
When it comes to comparisons, the word "di" can be translated as "than".
Let's take again the example given in lesson 4.7:
Lei corre | più veloce | di | te.
She runs | faster (more fast) | than | you.
On the other hand, the terms delle/dei/degli etc. are used when you have a definite article before the noun you are comparing the subject to (i.e. than the)
First, let's quickly look at the set of "compound prepositions" below in order to fully understand:
di + il = del
di + lo = dello
di + la = della
di + l' = dell'
di + i = dei
di + gli = degli
di + le = delle
di is a simple preposition, il/lo/la/l'/i/gli/le are definite articles. When we combine them, they become compound prepositions. Here is a very useful thread with more details about this topic: https://members.rocketlanguages.com/members/forum/italian-grammar/compound-prepositions#forum-post-41174
When it comes to comparisons, which compound preposition you choose (from the list above) depends on the definite article required by the noun in the sentence.
La tartaruga è più lenta della lepre.
The turtle is slower than the hare.
In this example "della" is a compound preposition (di + la). This is needed because "the hare" is translated as "la lepre" in Italian, and needs the definite article "la".
Let's see another example:
Carlo è più alto degli altri ragazzi.
Carlo is taller than the other boys.
In this example "degli" is a compound preposition (di + gli). This is needed because "the other boys" is translated as "gli altri ragazzi" in Italian, and needs the definite article "gli".
This applies to all other compound prepositions mentioned above.
To recap, "di" can be used as "than" when no definite article is needed before the noun (e.g. Io sono più intelligente di te / I am smarter than you). The other compound prepositions can be used as "than the" when a definite article is needed before the noun (e.g. Il telefono è più costoso del computer / The phone is more expensive than the computer).
Let me know if you would like to see more examples! :)