Yes, or something like it. Most accounts have him saying it to Halifax in 1940, though he possibly repeated it on other occasions. It is sometimes reported that he said "J'assume la France" rather than "Je suis la France". In either case, the expression signified his intention to assume responsibility for France.
Many of the smaller countries overrun by the nazis had properly constituted governments-in-exile in London. In some cases the monarch himself (or herself) had also fled to London. For nationals of these countries (those who wanted to fight on at least) the issue of legitimacy did not arise. The legal government continued to function, albeit from London.
None of this applied to De Gaulle's initially tiny organisation. For most people in 1940, the legal government of France was that of Petain, based in Vichy.
De Gaulle had no monarch, virtually no armed forces, no access to funds (beyond what the British allowed him) and no territory. For his project to succeed he had to create and encourage the idea that a legitimate French state existed beyond Vichy. He could only do so by projecting his personality and by acting - even from the tiniest beginnings - the role of the world statesman.