In the first stanza, the speaker thinks about a time in the past when she was a child and would have been scared of the stormy night that she is experiencing at that moment. The storm here could be a literal storm, or it could be a metaphor to represent the emotional turmoil of death. A storm is a good metaphor for death because a storm can be violent and chaotic and damaging, and death is likewise, from an emotional perspective at least, violent, chaotic, and damaging. If we take the storm as a metaphor, then the speaker may be reflecting upon her feelings as a child when her father died or was about to die.
In the second stanza, the speaker imagines what she might have said when faced with this storm as a child. At the end of the stanza, she imagines she would have lamented that
his road is bare of hedge or tree,
And he is getting old.
This may be a reference to her father. The father's "road," meaning his life, is reaching its end, and the father, "getting old," is close to death. This then seems to suggest that the speaker is reflecting on how she felt as a child when she knew that her father was dying. Our understanding of the storm as a metaphor for death can be refined a little here, and we might now read the storm as more specifically a metaphor representing perhaps the father's illness leading up to his death.
At the end of the poem, the speaker says that her father has since
won that storm-tight roof of hers
Which Earth grants all her kind.
The "storm-tight roof" here is the ground, and the meaning is that the father is now buried beneath the earth, in his grave. This is the fate which "Earth grants all her kind," meaning that all men must die and will, when they die, be taken back into the earth.
In the third and fourth stanzas, the speaker lists the various characteristics of the house she has inherited from her father. She describes, for example,
the fitful chimney-roar,
The drone of Thorncombe trees
and also the "clacking garden-hatch." The implication is that the house remains the same, as it was in her father's time, and this emphasizes the idea that time passes, life goes on, and death is the natural, inevitable end.