Paraphrase
Everything, insofar as it is in itself, endeavours to persist in its own being.
Everything endeavours to persist in its being
Claims
Interpretation
Individual things are modes that express God's power in a determinate manner.
Interpretation
No thing contains anything that could destroy it (III.P4), and it is opposed to all that could take away its existence (III.P5). Therefore, insofar as it can, it strives to persist.
In Plain Language
Here it is — the conatus, Spinoza's most celebrated concept. Every existing thing pushes back against its own destruction. Not because it "wants" to survive in some conscious sense, but because its very being is an expression of the power of substance. A rock resists shattering; a plant turns toward light; a person fights for breath. This is not a mysterious vital force added to things — it is what it means to exist at all. To be is to strive.
Why This Follows
From ce-05 (no internal destruction) and ce-06 (contraries expelled): since nothing in a thing tends toward its own annihilation, and it structurally resists what would destroy it, the thing's existence is itself an active endeavour to continue. Spinoza adds that things are modes expressing God's power (I.P25, I.P34), so their persistence is a direct expression of that power.
Conatus — the striving to persist in being — is the fundamental drive from which all emotion will be derived.
Connected Concepts
Is the conatus just biological survival instinct, or is Spinoza saying something more radical — that even non-living things "strive"? What difference does it make?