Finding 'Captains Log' - what does it do?

I think your flagship is a medium hulled survey craft – if you upgrade it to a warship, you get quite a lot of experience from this anomaly, which means more HPs. If you’ve collected a few of the +4hp anomalies too, your flagship can end up as a bit of a monster. I think I had one with 90hps that had never entered combat – I think a medium hull has 20 normally?

If memory serves, it gives you 3 experience each time you run into it.  Three or four of them are enough to get your flagship leveled up once or twice once you upgrade it to a low tech weaponized ship (experience required for next level is based on attack + defense total of ship in question), but in all honesty you're better off running around killing whatever the AI produces (as experience gained is dependent on attack + defense of ships killed).

The +4 hp anomalies are far more valuable, particularly before TA's hp modules are easily accessible or worth the space (also in DL/DA).