The stars are the monster's Level.
If you want to Normal Summon a monster of Lv5 or 6, then you must Tribute Summon it by tributing a monster from your side of the field. If the monster is Lv7 or more, you must tribute two monsters.
Levels have little effect in-game, they only have an effect when another effect specifically looks at them. So Burden of the Mighty looks at a monster's level to determine how much ATK loss it suffers.
The most important thing Level does is how it works with Synchro Summoning. To Synchro Summon a monster, generally you must send a Tuner, and one or more non-Tuners (some rare cases allow more than one Tuner, others require specific components) with combined level equal to the level of the Synchro Monster. So Lv8 Stardust Dragon could be summoned using Lv2 Krebons + Lv6 Destiny Hero Malicious.
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