Q:

How do we know how the Romans pronounced Latin?

A quick Google Search says plenty of things about Roman Latin pronunciation, and since it's an edu domain I'm inclined to believe it. However, the closest to citing a source it gets is saying "we know from grammarians"; it doesn't say anything specific. 
How do we know how Romans pronounced Latin? 

A:

A standard work in this area is Vox Latina, by W. Sidney Allen.  The author answers your question in his foreword, identifying 6 types of evidence:

specific statements of Latin grammarians and other authors
  regarding the pronunciation of the language; 
puns, plays on words, ancient etymologies, and imitations of natural sounds;
the representation of Latin words in other languages;
developments in the Romance languages; 
the spelling conventions of Latin, and particularly scribal or epigraphic
  variations; and 
the internal structure of the Latin language itself, including its metrical patterns.

The evidence for some sounds, of course, will be stronger than for others:

The degree of accuracy, with which we can reconstruct the ancient pronunciation varies from sound to sound, but for the most part can be determined within quite narrow limits.

Briefly, here are examples of each of these six points from Allen's work:

Latin grammarians.  Prominent rhetoricians and grammarians instructed their students in proper pronunciation.  For example, Cicero described the pronunciation of the diagraph ch, and Priscian and Velius Longus attempted to explain the sound of the final -m.
Plays on words.  Cicero gives a frequently cited example of this, providing evidence that v was pronounced as [w], when he equates the sounds of Cauneas and Caue ne eas, which would only make sense if the u of caue was similar to that of Cauneas.
Word borrowing.  Latin shared vocabulary with Greek and other languages, with both languages borrowing from each other.  The letters used in transcriptions of these words give clues into the Latin sounds.  For example, evidence for n being pronounced as [ŋ] before certain consonants comes from transcriptions from Latin to Greek and vice versa.
Romance languages.  The development of Romance languages like Spanish from Vulgar Latin is sometimes helpful with respect to pronunciation.  For example, the relative lack of aspiration in voiceless stops (p, t, c, qu) is suggested by the lack of aspiration of these stops in Romance languages.
Spelling.  The conventions and variations of Latin spelling provide helpful pronunciation clues in a number of cases.  For example, thanks to widespread spelling confusion, we can trace changes in the pronunciation of the u-consonant back to the first century.  Similarly, inscriptional evidence is used to establish the length of vowels before “gn”?.
Structure and meter.  The meter of Latin poetry is particularly helpful in ascertaining the lengths of vowels, such as in determining the correct pronunciation of the names of the letters of the alphabet.

A:

An important source of information is comparison to other languages.
For example, Cicero was spelled as Κικέρων1 in Greek.
If we believe that the Greek kappa was pronounced as /k/ rather than /s/ or /ts/ or anything else, we can be confident that c was pronounced as /k/ in Latin as well — at least in this name.
Similarly, the name Caesar gave rise to the present German word Kaiser.
The loan was, as far as I know, made in the ancient times.
If we know something about how the pronunciation of German(ic languages) evolved, we know something more about how Caesar pronounced his name.

1
Thanks to chirlu (see comments) for the correct Greek accent and the final nu.
I don't know if the final nu was used in all occasions, but that is besides the point.

