Why do we use:
- to start laughing
- to start to laugh
And what is the construction verb + (verb + ing) called?
To answer Why is it "to start laughing" and not "to start to laugh"?, I will point out that both alternatives are equally possible (but there are some minor points we should be aware of) by quoting a sub-entry from Practical English Usage by Michael Swan. (Its main entry is about some verbs and adjectives that can be followed by either -ing forms or infinitives.)
299.10 begin and start
Begin and start can be followed by infinitives or -ing forms. Usually there is no important difference.
She began playing / to play the guitar when she was six.
He started talking / to talk about golf, but everybody went out of the room.
After progressive forms of begin and start, infinitives are preferred.
I'm beginning to learn karate. (NOT
I'm beginning learning karate.)
Infinitives are also preferred with understand, realise and know.
I slowly began to understand how she felt. (NOT ...
He started to realise that if you wanted to eat you had to work. (NOT ...
To answer What is the construction 'verb + (verb + ing)' called?, I will base my answer on chapter 13 section 4 (pages 214-6) of A Student's Introduction to English Grammar by Rodney Huddleston, Geoffrey K. Pullum, and answer that this construction is a catenative construction where the matrix verb (i.e. the main verb) is a catenative verb, and the complement is a catenative complement.
In our case, the catenative verb is start and the catenative complement of to start to laugh is a to-infinitival clause, and the catenative complement of to start laughing is a gerund-participial clause. (Please refer to section 3 in chapter 13 for more details on "the functions of non-finite clauses".)
According to the book, the term 'catenative' is derived from the Latin word for "chain". To keep this answer short, I provide a link to page 215 on Google Books, along with a screenshot below. The page has a very good example of such a 'catenative' chain.
To start is one of those verbs that can be followed by both gerund ([verb]-ing) and infinitive (to + [verb]). There is no difference in meaning whatsoever. This is also true for to begin and to continue.
That said, bear in mind that in continuous tenses the infinitive is used, i.e. "is starting to do something".
Here in started laughing : 'laughing' is a gerund.
To make it more clear a gerund is nothing but 'Verb+ing' form which is a noun.
Answer to your second question.
Verb+(Verb+ing) is nothing but verb+gerund --> it is nothing but verb+noun.
Here are the example that clears your confusion.
He likes playing cricket.
He hates smoking.
He masters gokarting.
Sometimes two expressions are used side by side. 1 to start with laughing ("with" was dropped later on) and 2 to start to laugh.