You see this a lot on formal government and institutional documents such as declarations and diplomas. What, exactly, is being said?
It is a very formal (or dated) version of “to whom it may concern”. You write it in particular on letters patent, such as royal, presidential, diplomatic proclamations. Because these are not sent to a named recipient, but open letters intended to proclaim or attest of something, and thus be presented to the public at large. As you noted, they are also used in diplomas and academic degrees.
F'x is correct in the main, but it is useful to note that these presents is a legal term that makes specific reference to the document in which the words being read are contained.
As NOAD gives it:
these presents formal Law — this document : the premises outlined in red on the Plan annexed to these presents.
And the etymology:
ORIGIN Middle English : via Old French from Latin praesent- ‘being at hand,’ present participle of praeesse, from prae ‘before’ + esse ‘be.’