How should I keep date palm fruit at my home?
The dates you have are probably brahi, whereas the ones you remember eating before are medjhoul. These are different types of dates, and have slightly different flavour profiles.

As for storage, they should be fine for a few days at room temperature. For anything longer than that, they should be kept in the fridge, particularly if you live somewhere hot/humid.
I also vaguely remember an old Israeli habit of eating yellow dates frozen, or after a few minutes out of the freezer.