Identify the participial phrase and the word it modifies. seizing the weapon from the intruder, he ran quickly from the house. participial phrase: word modified:

Participial phrase:Seizing the weapon from the intruder

Word modified:he

The participial phrase here is seizing the weapon from the intruder. 
Participles look like verbs, and they are formed by adding -ing to the verb (for example, eating, sleeping, seizing, in this case, etc). They look the same as gerunds - however, whereas gerunds are used as nouns, participles are used as either adjectives, or adverbs. Here, the phrase is used as an adverb.

Given that the phrase is used as an adverb here, the word it modifies is ran. 
Adverbs modify verbs, meaning that they explain them in more details. Here, you can ask the question - How did he run quickly? And the answer would be - Seizing the weapon from the intruder. The phrase explains the verb.