What this means that "bitcoin wallet software" uses SPV, and the bitcoin network, and this is not a full-node software ?
It means that the wallet software is using
Simple
Payment
Verification and therefore does not require to download the whole blockchain to verify that transactions are included in a block.
You can read up on that in the bitcoin whitepaper or on one of those dozens websites explaining SPV.
and why it start to synchronizing with network and says 6 peers connected for example,
Since it isn't full-node why it start to connecting to peer?
Because a SPV wallet still needs to download the block headers (= synchronization).
To accomplish that, it connects to other peers.
Maybe i miss-understand the concept behind bitcoin!
Maybe.
If you tell us what you believe to know, we can point you into the right direction.