But what surprises me, there is no one who can prove that the private keys are safe?
There must be someone who can explain that and why the keys are secure.
There has to be a fact that exists or not.
Where is the explanation?
I can try to explain how it works...
Replace BTC with LTC or DOGE throughout, because it applies to them just the same... for the sake of simplicity I'll assume you're importing a BTC private key.
When the initial distribution was made, the CLAM developers looked at the BTC blockchain, and made a list of all the addresses that had funds. They converted each BTC address into the corresponding CLAM address. The only difference is the "version number" that is used to convert the RIPE-MD hash into an address - in BTC it's version 1 (which makes BTC addresses begin with a '1') and in CLAM it's version 137 (which makes CLAM addresses begin with an 'x'). Then they funded each of the corresponding CLAM addresses with 4.6 CLAMs.
To spend those 4.6 CLAM outputs, you need the addresses' private keys. The private key for each CLAM address is the same as the corresponding BTC address' private key. To spend an output, you create a transaction that moves the value to a new address, and sign it using the private key. Then you broadcast the signed transaction to the network. The private key itself is only used for signing and isn't broadcast.
Whether "the keys are secure" or not depends on whether the CLAM client software is trustworthy or not. You are giving it your private keys, and trusting it to only use them to sign a transaction that moves your 4.6 CLAMs for you. In theory it is possible that it could also check whether there is a balance at the corresponding BTC address. It could, in theory steal any such BTC. There have been no reports ever of the CLAM client software stealing anyone's BTC, so it's likely safe. Also the source code is open source. You can inspect it, and build the client for yourself from source. That's what I do.
Incidentally, note that any random shitcoin wallet you download is just as able to steal your BTC. When you download and run a program, it can do whatever it wants on your computer. There's nothing to stop it looking in your bitcoin folder and uploading your BTC private keys to its author. There's nothing to stop it watching you type your wallet passphrase into the Bitcoin wallet, and then telling the passphrase to its owner. The message here is to be careful *whenever* you download and run programs from the Internet. Any program is potentially coin-stealing malware.
This probably isn't what you were hoping to hear, but it's the truth. Downloading and running programs basically allows the program's author to do whatever he likes on your computer, whether or not that program has an "import wallet' menu entry.
Personally I trust the CLAM client, and have imported wallets which have hundreds of BTC still in them without any issue.