I always learnt it was pronounced the same as how 'e' is usually pronounced in German (in either its short or long forms respectively). But then the question is: why have a different letter for it? Surely the original sound must have been different?
“Originally” is a problematic concept. The letter “ä” was not used in Old and Middle High German. The plural of gast is gesti in OHG and geste in MHG. In early New High German the letters ä, ö and ü (or rather: a, o, u with a small superscript “e”, but I cannot find these in Unicode) were used to indicate the umlauted forms of a, o and u, but the distinction between e and ä was dictated by etymology and not by pronunciation; ä was used where there are obvious cognates with a, as in Gast > Gäste. The spoken distinction between “e” /e/ and “ä” /ɛ/ is essentially a spelling pronunciation adopted in the standard “Bühnenaussprache”.
Reading Siebs (19th printing, 2000—this edition introduced the notion of a relaxed gemäßigte Hochlautung) again, here are some facts about High German:
- In Standard German, there is no distinction between short e and ä, they are usually pronounced /ɛ/
- In Standard German, there is a distinction between long e and ä, the former is pronounced /e:/ and the latter /ɛ:/
- Those two prescriptions are also in effect for gemäßigte Hochlautung
- Pronuncing long ä as /e:/ is non-standard
Siebs also makes a historical remark about German orthography: German orthography uses the letter ä when there is a related word with a and e otherwise, disregarding historical development and dialectal pronunciation. In fact, at least some German dialects make even more distinctions on the e-sounds, depending on when the umlaut happened (Primärumlaut and Sekundärumlaut). There are also words spelled with a long e that are dialectally ä words (e.g., leben "to live").
Until rather recently, ä represented the sound /ɛ/, also called "epsilon". This is similar to the sound of e (conveniently written /e/), but with the tongue slightly lower: it's the vowel in English "pet". Some German-speakers still use this pronunciation, but only when they're speaking carefully—that is, it takes conscious effort to get it "right". In casual everyday speech, it's merged into /e/ over the last century.
It's not clear what the "original" sound was, but we can make a guess. The process of front-umlaut in German (and other Germanic languages) involved shifting vowels forward in the mouth in certain contexts. ö /ø/ is o /o/ shifted forward, for instance, and the same for ü /y/ from u /u/.
So based on this evidence, we could say that the "original" underlying phoneme for ä might have been /æ/, a low front vowel (the sound in General American "cat"). But this is nothing but an educated guess. The difference between the actual sounds [æ] and [ɛ] is rather small, and I don't know of any evidence that would tell us conclusively that it was originally pronounced [æ].
As far as modern Standard German (Hochdeutsch) is concerned, (1) 'ä' is merely an orthographic device whose sound value is the same as 'e' in the same position; (2) there is some vacillation among speakers when it comes to the long vowel; the short variant is always [ɛ] unless someone wants to sound comical.
Standard pronunciation suggests to read long 'ä' as [ɛ:] as contrasted with long 'e' [e:], but this is not generally followed except for a few isolated cases like the interjection 'äh' [ʔɛ:] (people often perform sounds in interjections that they'd otherwise claim not to say at all, like the click in E. 'tsk').
The reason standard pronunciation wants us to read 'ä' as [ɛ:] at all comes from the misunderstanding that where there's a separate way to write words, there should also be a separate way to read those words. Needless to say, this idea fails in many places, not least of which is the contrast of short 'ä' and 'e', which are undisputedly always [ɛ]. There's also some utility to keeping [e:] ~ [ɛ:] separate as there are a few minimal pairs that would otherwise sound the same, such as Beeren ~ Bären (berries, bears) and Ehre ~ Ähre (honor, ear of corn).
So why do people not follow and keep [e:] ~ [ɛ:] separate? One, because it is artificially introduced, phonetically not very salient and, usage-wise, not very useful distinction to be made, as only a handful of lexical items profit from the distinction. Also, there's a constant interference from the many dialects that have only the one or the other vowel, or, when they have both, use them in ways that conflict with what the standard would have us do. As in other languages, most people do not natively speak the perfect standard natively; rather, it is an acquired dialect that is approximated from the sound-base of what one grew up talking.
One reason why one should definitely not follow the folly and separate [e:] ~ [ɛ:] is because it's at odds with the overall German vowel system. German has 7 simple full vowels, viz. /a,e,i,o,u,ø,y/ (plus schwa), all of which come in a short and a long variant. The short ones are generally laxer / more open than the tenser / more closed long ones, so [a, ɛ, ɪ, ɔ, ʊ, œ, ʏ] contrast with [a:, e:, i:, o:, u:, ø:, y:]. An extra [ɛ:] does not fit well into this symmetry.
(There's yet another variant for each vowel that is used mainly in words of Latin or Greek origin; these are described as being a mixture of the short and the long ones in that the vowel quality is rather tense, but the quantity is short or half-long, which gives [aˑ, eˑ, iˑ, oˑ, uˑ, øˑ, yˑ]).
As for why an extra letter Ää was introduced into the spelling when it doesn't correspond to a sound of its own, well, that's a good question. On the one hand, using 'ä' to mark the Umlaut does tie related words together, so you can immediately see that 'Korn', 'Körner', 'Fluss', 'Flüsse' and 'Lachen', 'Lächeln' are related. OTOH we write a lot of words with 'ö' and 'ü' that do not have a modern counterpart with 'ä', 'o', 'u', ex. 'Öl' (*Ol), 'Süden' (*Suden), 'Ähre' (*Ahre). Worse, there are lots of historical Umlaut vowels that are not marked in the orthography at all, ex. 'Gewitter' (<Wetter), 'Gebirge' (<Berg), and then there's a fair number of high-frequency words where usage has been non-uniform over the past one or two hundred years, ex. 'ächt'-'echt', 'Ältern'-'Eltern', 'Stängel'-'Stengel', 'enbläuen'-'einbleuen'.