The comments have it right. It means a dream that has come true and "that has" can be omitted. This works generally for several specific verbs (run, gone, turned). For some constructions it doesn't work:
A man that has become king
Cannot be rephrased to:
*A man become king
Run only works in certain usages, such as run amok or run rampant. Even these usages might be a little questionable.
The athlete that has run a mile
Cannot be rephrased to:
*The athlete run a mile
Additionally, a dream come true is itself a very common idiomatic expression which is rarely phrased any other way (unless the author is intentionally avoiding the idiom).
Also note that this is similar- but not the same as -the omission of that has been. For example,
A man that has been crowned king
Can be rephrased as
A man crowned king
This construction, unlike the one in the original question, works for far more verbs and is far less idiomatic.