Why do nodes trust other nodes to approve their transactions? I know that the blockchain technology works so well, because everybody has basically a copy of the last ledger.
Therefore, if a transaction happens, everybody inside the network has to &quot;approve&quot; the transaction to do it.
These other participants are then seen as trusted sources.
However, what makes them into trusted sources? Let's say there are only three participants, two make a transaction and one has to decide if the transaction is acceptable or not (does the clearing). What makes him a trusted source?
Are these thoughts right or where am I wrong?
I appreciate your replies!
I know that the blockchain technologie works so well, because
  everybody has basically a copy of the last ledger.


Yes, every full bitcoin node has a copy of the blockchain (aka ledger).


  Therefore, if a transaction happens, everybody inside the network has
  to "approve" the transaction to do it.


Correct, every transaction is propagated to all bitcoin nodes and it is validated by each one of them. Some of those nodes also do mining and they will include this transaction to a block. If they mine the block successfully (before any others mine their blocks successfully) they propagate their block to all the other bitcoin nodes which validate the block and all the transactions within. Thus, transactions are validated a second time.


  These other participants are then seen as trusted sources.
  
  However, what makes them to trusted sources. Let`s say there are only
  three participants, two make a transaction and one has to decide if
  the transaction is ok or not(does the clearing). What makes him a
  trusted source?


No one is considered a trusted source. Bitcoin enables trustless transactions. If we have 1000 bitcoin nodes in the network and user A makes a transaction that sends some coins to user B all 1000 bitcoin nodes will validate the transaction. Then, when the mined block is propagated all 1000 nodes will validate it again. Note, that users A and B may or may not run a full bitcoin node themselves.

Hopefully, that clarifies things a bit.