Christianity, especially in its infancy was quite vulnerable to those hostile to its doctrines. Christians were often martyred and persecuted for their beliefs. Also remember that Christianity began in Jerusalem, where Jesus had lived and ministered - if Christians were exaggerating or distorting the truth of what had just happened, you can be sure non-Christians would have immediately pointed those inaccuracies out. But we don't have a record of those claiming Jesus did not do the things His followers said he did.

The gospels were also incredibly well preserved. There is a large multiplicity of the manuscripts, and within these manuscripts, there is a high degree of fidelity. And although there are variants between manuscripts, these variants are almost entirely inconsequential (ie spelling) and none of them affect any major Christian doctrines.

Finally, the gospels, like all other books of the Bible, are all markedly different and yet are consistent with one another. This shows a high likelihood of independent and authentic observation of the same event.

See also: Are the gospels reliable?