1) By examining the use of rhyme in verse.
Fear no more the frown of the great
Thou art past the tyrant´s stroke
Care no more to clothe and eat
To thee the reed is as the oak.
This shows that for Shakespeare the word eat was pronounced with the same vowel as is the word great today, i.e. as /e:/.
2) By comparing the forms of loan-words with their originals
Old English wine from Latin vinum /winum/ has an initial /w/ as did Latin at the time of the borrowing (in common West Germanic, on the continent) Later, the word was re-borrowed with the pronunciation /vi:n/ to give vine which in Modern English means the bush on which the wine grapes grow. The latter pronunciation is due to the fact that in Latin an original /w/ had developed to /v/ between the periods when the two words were borrowed.
3) By observing reflexes of older pronunciation in the present-day language (this method is termed internal reconstruction).
For instance, linguists assume that in Old English the fricatives /f,s,θ/ were pronounced voiced in intervocalic position.The justification for this assumption derives from such alternations as hoof : hooves in Modern English. The voiced fricative in the plural is due to the fact that originally this was in intervocalic position, cf. Old English hofas. Analogical change may, however, render the former distribution invisible. For instance, the word roof frequently has a plural with a voiceless fricative - roofs - although originally it also had a voiced fricative deriving from an intervocalic position.
Note. Analogical change is where a minority form is made to conform to majority usage in morphology. For instance in Middle English the word eye had a plural in -n, cf. German das Auge : die Augen, but later it was made to conform to the vast majority of words and adopted an -s plural. Only those words which are very common in a language - i.e. at the core of the vocabulary - are resistant to such regularisation, e.g. the irregular plural in English which show a reflex of former umlaut, cf. foot : feet, man : man (parallel to German Fuß : Füße, Man : Männer).
4) By examining related forms of words from other dialects or languages (this is the comparative method). For instance the sequence /wh/ was formerly pronounced /ʍ/ although it is currently /w/.
However, if one looks at conservative dialects like Irish or Scottish English one finds that the older pronunciation is still present. Across languages one can find supportive evidence of assumptions about pronunciation as well. Take for instance the word kyning in Old English. Linguists assume that the y refers to a front rounded vowel. Now Modern English has /ɪ/ in this word, but the German word König shows a front rounded vowel (albeit mid rather than high). Furthermore, we know that Germanic loan-words in Finnish are very old, and the form kuningas exists there for `king´. This helps us also because linguists assume that there was a later phenomenon of umlaut whereby back vowels were fronted when a high vowel was present in a following syllable (as is the case here).
5) By considering occasional `incorrect´ spellings which betray that a sound change had taken place.
For example, a well-known change in Middle English in the south of England is the loss of the voiceless velar fricative in words like night , now /nait/ formerly /ni:t/ from a much earlier /nɪçt/. Now there are spellings from the 14th century like wright for write. This shows that the /x/ must have been lost in the scribes dialect as the word write was always pronounced with a long /i:/ and never had a /x/ in it.
6) By considering prescriptive comments on contemporary language use. This is not a great help in Old English as there are not many such comments available. But in the Middle English, and certainly in the Early Modern English period, such comments are a valuable source of information on pronunciation. For instance we know that from contemporary remarks that the high back vowel in but /bʊt/ was lowered to the unrounded low vowel during the 17th century in the south of England giving us the present-day pronunciation /bät/. Similar comments can help us to date the loss of /r/ at the end of syllables, again in southern British English, a development which is responsible for the modern Received Pronunciation seen in words like car /kɑ:/ and card /kɑ:d/.
7) Very generally, by applying one´s knowledge of what is likely in language change and historical linguistics.
A normal development in many languages is the palatalisation of velar stops. There is a progression to this change which starts with /k/ and leads to /ʃ/ via /tʃ/. Thus when one takes Latin camera `room´ as a starting point and observes Modern French chambre with an initial /ʃ/ then one is justified in assuming that inbetween, in Old French, the pronunciation /tʃ/ must have existed. Indeed this is corroborated by the English word chamber which was borrowed from Old French (in the Middle English period) and which shows the pronunciation /tʃ/ which one would expects from this intermediary stage between Latin and Modern French.
The following book contains much information on how linguists reconstruct previous stages of language (here: Old English). The second item contains similar information but is more general in its scope.
Lass, Roger 1994. Old English. A historical linguistic companion. Cambridge: University Press.
Lass, Roger 1987. The shape of English. Structure and history. London: Dent.
The study of Indo-European is quintessentially about reconstruction as it is concerned with the earliest stage of the language from which all later ones are derived.
Baldi, Philip (ed.) 1991. Patterns of Change, Change of Patterns. Linguistic Change and Reconstruction Methodology. Berlin: Mouton de Gruyter.
Clackson, James 2007. Indo-European Linguistics. An Introduction. Cambridge: University Press.
Fortson, Benjamin W. 2004. Indo-European language and culture. An introduction. Oxford: Blackwell.
Mallory, J. P. and D. Q. Adams 2006. The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World. Oxford: University Press.