The skylark has a beautiful song, but as with all birdsong, it has a very important purpose. In this particular case, it's to let the babies in the nest know that their mother is near.
In music a “strain” is a series of phrases that go up to make a melody. The skylark's strain in Wordsworth's poem is “love-prompted,” meaning that it is motivated by love. In saying this, the poet sentimentalizes the bond between the skylark and her nestlings. What to the scientist would be simply a fact of nature, becomes, in the poet's hands, a bond of love and affection.
The skylark's transcendence of the merely factual is reinforced later on in the second stanza:
Yet might'st thou seem, proud privilege! to sing
All independent of the leafy Spring.
There's clearly something special about the skylark's song, something that sets it apart from the seasons of the year. She sings all year around, whether or not it's spring. In doing so, she “thrills not the less the bosom of the plain,” meaning that her glorious song brings joy and pleasure to the world below.
The “love-prompted” strain doesn't just reinforce a bond of affection between the skylark and her nestlings, it also established a connection between herself and the people who thrill to hear her wondrous song.