To understand what a participial phrase is we need to know what a participle is.  A participle is a verb which functions as an adjective. Participles come in two tenses – the present participle end with –ing, while the past participle is the past form of a verb if the verb had ‘have’ in front of it.  Therefore it usually ends in –ed  or –en – usually.  There are many exceptions of endings for the past participle as well, such as driven, led, swum, etc. (there are many lists on the web of  the past participle tense: irregular verbs most commonly used.)
A participial phrase appears in relation to the participle in a sentence.  The participle, plus any complements and modifiers or predicate words of the participle, is called a participial phrase.
The participial phrase thus acts as an adjective and it follows the noun that it modifies or precedes the noun it modifies, with the noun immediately following the participial phrase.
The participial phrase contains the verbal, combined with a prepositional phrase, and follows the noun it modifies.  The participial phrase serves as an adjective.

examples of participial phrase usage

with present participle:
The audience clapping with enthusiasm did not hear the storm outside.
The bushfires spreading across the land could not be contained.
The fishermen approached the river teaming with fish.

Past participles in participial phrases:
Tired and eaten alive by insects, the travelers reached a small village.
The burglar, caught unawares, made a dash for it.

A participial phrase appears in relation to the participle in a sentence.  The participle, plus any complements and modifiers or predicate words of the participle, is called a participial phrase.