It is not accurate to say that the Greek alphabet developed from the Hebrew alphabet as we know it. Instead, the two have a common predecessor in the Phoenician alphabet. In this sense you could say that the common predecessor of the Greek letter alpha (α) and Hebrew 'aleph (א) was a Phoenician alphabet letter ʾālep (𐤀). When I transliterate the name of the letter ʾālep, please note the initial apostrophe, which transliterates an ʾālep letter. It's unsurprising that the name of a letter begins with that letter.
All letters in the Phoenician alphabet represent consonants. The Phoenician alphabet was used to write a number of languages, including Phoenician and Hebrew.
In Phoenician, ʾālep is never silent. It represents a glottal stop, a sound denoted as /ʔ/.
Glottis sits in your neck, it is the opening between your vocal cords; it's what "gives you your voice". The best known use of the glottis is forcing your vocal cords to vibrate (i.e., to interrupt the flow of air about a hundred times per second). Based on what is happening above your neck, this produces various vowels or it gives voicing to any voiced consonants. However, it is also possible - and rather common - to to close the glottis briefly (stopping the flow of the air through the neck momentarily), and then release it either on the onset of a vowel (which produces a glottal stop, a consonant articulated purely at the glottis), or perhaps together with articulating a consonant somewhere else (which produces a particular "glottalized consonant".)
A glottal stop isn't the only way to start articulating a vowel. A glottal stop gives you an abruptly starting vowel, a vowel which starts with a tiny "explosion". You could also keep your vocal cords somewhat relaxed (but not too relaxed or too distanced from each other which would disable vibration altogether) and then the vowel will begin sounding gradually.
In Phoenician, ʾālep is always pronounced as a glottal stop.
In (Biblical or older) Hebrew, 'aleph can be pronounced as a glottal stop or it can be completely silent.
In (Ancient) Greek or (Modern) English, a word-initial vowel can start with or without a glottal stop. That's somewhat similar to Hebrew, except that there's no tradition to have a letter for a glottal stop in Indo-European scripts, whereas Semitic scripts do have a letter which tends to have that function.
The previous paragraph is very possibly just a rather long lasting and geographically extensive artifact of the way how ancient Greeks chose to (re)interpret the Phoenician letter 𐤀.
How could that happen? Well, if you ask someone to pronounce a bare consonant for you, they can't. "Con-sonant" means the consonant needs to sit on a vowel somehow. In many languages, the "default" vowel that a speaker would use in this situation is traditionally /a/. So, when asked to pronounce /'/, the speaker says /'a/. And the listener may well choose to hear just /a/. It's also possible to imagine the opposite situation when you are trying to transcribe Greek names inside Phoenician texts and run into the problem that Phoenician words never begin with vowels while some Greek words do; it is tempting to hear and write an ʾālep in that situation.
Add to it a different structure of Semitic and Indo-European languages. Semitic languages can't ignore the equivalent of the glottal stop even in the languages where actually pronouncing it is optional, or where it is completely silent, because it has its place in (say) tri-consonantal lexical roots. Greek is grammatically more like English - the root of the word tends to stick together and so it doesn't matter very much how you choose to count its sounds.