As you correctly pointed out, many 一段 verbs have an older 四段 version. Many are formed by combining them with 在る, and 得(う)る, and 為(す)る, whose classical sentence-ending ("dictionary" or 終止形) form is only す.
This book, available online, explains it very well. (Unless you are a professional linguists, the book does a good job at making sense of and shedding light upon, as the title suggests, the history and development of the Japanese language. And it's free.) See pages 199-201, as well as those pages mentioned below. The following is a short summary.
You should know about bases (eg 未然形, see page 90), as well as about 四段・二段・一段活用(page 129) verbs.
There are three categories how (in)transitive verbs are derived:
(a) original intransitive verb --> transitive form
立つ→立つる (which has become たてる)
進む→進むる (which has become すすめる)
(b) original transitive verb --> intransitive form
(c) original and often archaic verb (possibly both trans. and intrans.) --> transitive and intransitive verb
出(い)づ → 出だす(trans.) and 出づる(intrans.)(now 出(い)でる)
定(さだ)む → 定むる(trans.)(now 定める) and 定まる(intrans.)
As you can see, the sound ~uru to -eru was very common, and it may have been influenced by the abolition of 二段 verbs. Formerly, you would talk about 子食ぶ(the child eats) and 食ぶる子(eating child), but 食べぬ子(child which does not eat). People stopped changing the vowel accordingly and picked one vowel for all bases.
While this processes is about transitivity, it explains how some older verbs have turned into their modern form. 食ぶ→食べる, 落つ→落ちる, 得(う)る→得(え)る, 飢(う)う→飢える.
(However, see page 151 and following of above mentioned book about the origin of the different types of conjugation, where the author speculates all 二段 and 一段 verbs were obtained via agglutination, eg by adding verbs.)
In fact, the original conjugation of 得(う) was 終止=う, 連体=うる, 連用=え, and when a word such as 進むる consists of 進む+得る, it is easy to see how it could have become 進める.
The two examples from (c) make the derivation particularly clear:
出だす is いづ+す. Note this is the same way how causative verbs are formed. (For example, the causative in classical Japanese of 読む is 読ます, whose 連体形 is 読まする, now changed to 進める.) So 出だす is lit. "make come out", ie "to put out". See pages 164-173 of the above mentioned book for more details on causative verbs.
定まる is 定む+在る, lit. "be decided", and thus intransitive. You could draw a parallel with the modern construction 書く→書いてある ("be written").
定むる is 定む+得る. Note how this is similar to the classical passive/potential form. (For example, the passive of 読む would be 読まる, 連用形=読むるる, now changed to 読まれる, which may contain an additional 在る, yomu+aru+uru to account for the a-vowel.) See pages 156 and the following of the above mentioned book for more information on passive/potential verbs.
Literally, 定むる would be "get [to make] a decision", and thus transitive. Note that 得(う) can mean 身に受ける・身に付ける・手に入れる・自分の物にする as well as ~する事が出来る. (全訳国語辞典) You could draw a parallel with potential verbs today, such as 読める, which is 読む+得(え)る.
On a side note, this compounding explains why you can often tell whether an -iru/-eru verb is 一段 or 五段. 送り仮名 tends to be used for the inflectional part that changes. 一段 verbs contain more okurigana and syllables that don't change; such as さだむ, which is 定む, when we add ある, it becomes 定まる. Okurigana 五段 verbs are only the syllable which changes; such as 滑る being 五段, and 統べる being 一段 deriving from 統ぶ. The same can be said for 食べる, where the べ-syllable was originally part of the inflectional part of the 下二段 verb 食ぶ.
This ended up longer that I had expected. This question is a good example of where the answer to one questions raises several more.
Referring to a comment, and incorporating some information from a book I read after I wrote this answer originally, a few more thoughts concerning the disappearance of the bigrade conjugation and its etymology:
When most bigrade verbs shifted to unigrade during the transition to Modern Japanese (1600-), the alternative vowel was already part of the system. Bigrade verbs are found in Old Japanese already, and one inquire as ti their etymology. See "A history of the Japanese language" by Frellesvig (Cambridge, 2010, 1st ed.), page 99-101 and 391-392. In short, he suggest phonological changes, in particular vowel and consonant deletion, so that ake- (to open) formed its conclusive (終止) (like 聞く = kik- + u) via akeu->aku, and 起きる:okwi+u>okwu>oku. He then suggests that bigrade verbs form the adnominal (連体) and exclamatory(已然) with the conclusive(終止) as a base, so that akuru=aku+ru, okuru=oku+ru. Quadrigrades use the base: kak+ru=kaku (consonant deletion), 已然形 analogous. When the 終止形 fell out of use, he argues, akuru and akure were not based upon this conclusive anymore, but derived from the stem akuru=aku+ru. Thus, change took place so that both the 連体 and 已然 were now derived from the basic stem ake- and oki- as well.
This does not necessitate speakers to have been aware that akuru<-aku+ru<-、ake+u, they might have simply noticed the resemblance akuru=aku+ru and akure=aku+re (ie the first "<"), which was not possible anymore once aku had come out of use. ake- still existed, and become a substitute. Incidentally, for some time both forms existed parallel to each other: fe -> furu/feru (経る)
These deletions are not hand tailored to fit this purpose, they occur in several unrelated words in Old Japanese as well, eg: wa+ga+ipye->wagapye (我が家), myesi+age->myesage (召し上げ), toko+ipa->tokipa (常盤), and interestingly, ko+i->ki (き, 連用 of 来る) as well as inflectional forms such as kakite+ari->kakitari (書きたり), arazu+ari->arazari (非ざり)
See page 39 of Frellesvig.
One [vowel] is elided. The second vowel is only elided when a monosyllabic morpheme is followed by a vowel initial polysyllabic morpheme, elsewhere the first vowel is elided.