what is the purpose of a title sequence and film opening?

There is more than one reason to why a title sequence is used at the beginning of a movie.  A title sequence presents the title of the movie and the production and cast members.  A title sequence and film opening's main purpose is to set the tone of the movie early on.  This increases the audience's understanding of the movie.  For example in a title sequence/film opening, if a dark background is used or if a fast soundtrack is used, this sets a dark tone and suggests the movie will be a thriller or will be full of suspense.
 

For example as shown, the title sequence of the movie "se7en" sets a really dark tone.




A title sequence and film opening is supposed to suggest the genre/tone of the movie, however not give away too much about the movie itself.  For example, "the girl with the dragon tattoo" had a really good title sequence and film opening.  In the title sequence, a very fast soundtrack was used.  Also a very dark background was used, these two concepts combine to form a very dark tone.  A faded picture of a keyboard was shown which suggests there will be an investigation, but nothing about the investigation itself is given away.

A title sequence and a film opening is supposed to draw the audience into the movie, the reason for this is because a title sequence and a film opening is the first thing you see when watching a movie, after the production company is shown.  If the title sequence lifts the audience, it probably means the rest of the movie will.  Also a title sequence is not just there to interest the audience, it can also be used to give the audience some knowledge about who was involved in producing the movie and make the audience even more interested in the movie.  For example the cast/producer has their name written on the screen.  Someone who is a fan on David Fincher movies, will probably be happy to see his name written on the screen.