You need to equip fully enchanted weapon, armor and off-hand (bow). Fully enchanted means that the item should have 3 enchantment slots and they should all be upgraded to level 3.
You will start seeing items with 3 enchantment slots once you start playing in Adventure difficulty. As far as I can tell, Default difficulty does not spawn items of this kind. You can upgrade items by pressing Y on the item, pressing left/right to select the enchantment and then A to upgrade. Each upgrade requires enchantment points that you get for leveling up. Regular enchantment costs 1,2, and 3 respectively for a total of 6, while powerful enchantment costs 2,3 and 4 respectively for a total of 9 points to level up to 3.
What this means is that, you will need to level up 3 slots of 3 items to level 3. So, absolute minimum required enchantment points are 54, so you need to be at least level 55 to get this achievement. If powerful enchantments are included in the mix, you need more enchantment points. You can salvage items to get the points back, so do not be afraid to upgrade your equipments.
Edit: The enchantment in each of the slot may have to be unique and not same as the one in another slot. Thanks grrrrnade for this suggestion.
Edit 2: This may be patched at any time. All credit to xzDEATHWALKERzx for this.
With the latest update (09-08-20) that dropped at the same time as the Creeping Winter DLC they added Daily Trials. Some of these Trials give you a bonus enchantment.
I can confirm that these also count toward your total and count as a level 3 enchantment. I just popped the achievement as soon as I started the Trial. I was one enchantment short prior to this. Not sure if this will be patched but for now it seems to work.
For reference I had 3 weapon enchantments maxed, 2 armor enchantments maxed, and 3 bow enchantments maxed. The Daily Trial gave me a temporary Looting enchantment (a weapon enchantment and one I did not have already) and the achievement popped as soon as I started the Trial.
I guess the interesting takeaway is that the achievement calculates 9 total enchantments and not necessarily 3+3+3 for the requirement. In my case with the added temporary weapon enchantment this meant I had 4+2+3=9 achievement popped.