If i plug my power source into the PoE Midspan injector and than i connect Arduino to the PoE Midspan injector using an ethernet cable my Arduino doesn't starts.
If i plug my power source directly into Arduino it starts
If i plug an LED button into the PoE Midspan injector the button has power
Do i need a better power source for my Arduino (if yes what would you recommend) ?
The arduino PoE is REAL PoE, so you need an injector or Ethernet switch that actually supports 802.3af, not just a wiring hack that connects a power supply to the extra pairs...
He's using the Freetronics Arduino-like boards, which have a more flexible PoE setup.
From one of the Freetronics product pages:
Note that just like our Ethernet Shield with PoE support, the EtherMega provides a number of options for different Power over Ethernet methods. You can use the supplied jumpers and feed 7-28Vdc down the wire for cheap DIY version, or you can use our PoE Regulator 802.3AF along with a proper commercial PoE injector or switch that supplies 48Vdc. It's up to you.
See their tutorial (although the part about 802.3af is ... brief.)