What is the linguistic meaning of this sentence?
Vowels are always voiced, and have no attack of their own.
Attack in this context is the musical definition. Attack is how the sound/note is started. It is often paired with decay which is the diminishing of the sound into silence. Voicing is the sustained sound a vowel makes "ooooooooooooooooooo" when we push air from our lungs. They are claiming that vowel sounds can only be started by a consonant, putting its 'flavor' on the vowel sound.
Voiced: They require you to use your vocal chords.
Non-attacked: The do not require your tongue to make contact with the back or top of your mouth, other than the initial release of air when they are spoken individually or begin a word.
For example, say aloud "read." Your vocal chords will vibrate when you make the vowel sounds and your tongue doesn't hit the top or back of your mouth.
This applies to some consonants, too, such as the "R" and "S" sounds.