Q. What's the source and meaning of this verse?
Srimad Bhagavad Gita is the source of this verse:
अनन्याश्चिन्तयन्तो मां ये जनाः पर्युपासते।
तेषां नित्याभियुक्तानां योगक्षेमं वहाम्यहम्।।9.22।।
And here are some useful translations:
English Translation By Swami Adidevananda
9.22 There are those who, excluding all else, think of Me and worship Me, aspiring after eternal union with Me. Their prosperity and welfare (Yoga and Ksema) are looked after by Me.
English Translation By Swami Gambirananda
9.22 Those persons who, becoming non-different from Me and meditative, worship Me everywhere, for them, who are ever attached (to Me), I arrange for securing what they lack and preserving what they have.
English Translation By Swami Sivananda
9.22 For those men who worship Me alone, thinking of no other, for those ever-united, I secure what is not already possessed and preserve what they already possess.
For deep interpretation you may go with commentaries of Shankara, Ramanuja etc.
Now, taking about "how is it relevant with LIC" sounds partially off-topic or opinion-based. Though it is obvious that it should be analogous like:
BG: For those men who worship Me alone, thinking of no other, for those ever-united
LIC: Those who regularly pay instalments/premium!
BG: I secure what is not already possessed and preserve what they already possess.
LIC: We Secure the value of paid instalments/premium and equivalent value of your life after the life!