Q:

Can I say: "I'm the man who was there"? (changing from 1st to 3rd person)

In my language we can change the referring to a pronoun from 1st to 3rd simply by putting a relative pronouns, for example: 

"I am the man who was there"?

(I= 1st person. was=3rd person) 

"I am the man who takes it every day"

(I= 1st person. takes=3rd person)

"I am the man who writes here answers"

(I= 1st person. writes=3rd person)

"I am a man who likes to travel"

Are these type of sentences the same and correct in English? 
If they are, then what is the name for that kind of sentence structure?

A:

They are correct.
But they carry a lot of emotional and dramatic weight because of the "I am the man who" structure.
Scenario 1: I'm in an elevator, the power goes out for 5 minutes, I use the emergency phone to talk to the building manager, and then everything is ok.  When someone mentions it the next week, if I say "I am the man who was in the elevator", I would sound pompous and ridiculous.  The more normal expression would be "That was me."
Scenario 2: I'm an assistant on a dig in the Great Pyramid, and am the first person to see a relic that eventually proves something momentous about the ancient Egyptians.  30 years later, I overhear someone disputing that fact.  I respond by saying "I am the man who first uncovered the whatsis of Ramses: you, sir, are wrong!"  In that case, "I am the man" would be appropriate.
EDIT
Your added example of "I am a man who..." is less dramatic, although still not idiomatic.  People are more likely to say "I like to travel" or "I'm the kind of person who likes to travel".

A:

The sentences are correct. I would recommend not thinking of the switch to third person as being caused by the relative pronoun; rather, it's the noun phrase "the man" (the antecedent of "who" in your sentences) which licenses third-person singular agreement.

"I am the man who was there"

(This would be fine in any case, since "was" both the first and the third person singular form)

"I am the man who takes it every day"
"I am the man who writes here answers"

(the placement of "here" in this sentence is not correct)

"I am a man who likes to travel"

(These are all correct in terms of verb agreement because "a man who takes...," "a man who writes...", and "a man who likes..." are all correct)
Similarly, "you are the man who was..." or "you are the one who was..." are correct.
However, when the relative pronoun "who" comes directly after a personal pronoun such as "I" or "you", speakers generally prefer to use first or second person agreement. Note that this structure sounds rather formal and it may be inappropriate for ordinary speech.
Examples:

I, who am no great genius, have a weakness incidental to little
  ones. (Alciphron, Or The Minute Philosopher:
  In Seven Dialogues, by George Berkeley)

Now you, who are fast asleep with a broken crayon cleverly placed in
  your crib, have no clue as to what is going on... (Joy Robbers: Who
  Are They?, by Tanya M. Spiegel)

And if I still cannot find you, O God, then let me search my heart and know whether it is I who am blind rather than you who are hidden... (A Diary of Private Prayer, by John Baillie)

