Here's how it works:
The Pokemon with PickUp only has to be in your party, as said before. Every time you battle (wild Pokemon, Trainer, whatever), there will be a 10% chance that your Pokemon will "Pick Up" an item. It won't tell you, but the item will appear as the Pokemon's held item. If your PickUpper already has a held item, Pick Up won't work.
Level doesn't affect what items you can get in FR/LG, only in Emerald. Here's the list:
Found Often
Oran Berry (15%)
Found Sometimes
Cheri Berry (10%)
Chesto Berry (10%)
Pecha Berry (10%)
Rawst Berry (10%)
Aspear Berry (10%)
Persim Berry (10%)
Found Rarely
TM10 (5%)
PP Up (5%)
Rare Candy (5%)
Nugget (5%)
Found Very Rarely
Spelon Berry (1%)
Pamtre Berry (1%)
Watmel Berry (1%)
Durin Berry (1%)
Belue Berry (1%)