This is probably better for /r/linguistics than here, but I'll give a quick rundown.

First off, English is not heavily influenced by German, even in 'a very different form than today's German'. The first attestation we have of a language that could be called 'German' is Old High German, first attested around 700 AD, long after the Anglo-Saxons invaded and settled England. English is a Germanic language, and as such it has strong resemblances to German, but to say it's 'influenced by' German is like saying Spanish is 'influenced by' Romanian, when in fact they're of equal status on their family tree. Actual German influence (i.e. German attested beyond the first of OHG) is very little in English, mostly loanwords relating to arts and sciences.

French, meanwhile, is not a Germanic language and did heavily influence English; there is a huge number of French loanwords in English, mostly dating from the Middle English period, and English grammar went through a period of considerable simplification in the centuries following the Norman Conquest of 1066 (the Early Middle English phase). The ways in which grammar changed are a bit too complicated to go through here, but suffice it to say that a great deal of complexity in the endings of nouns (nominative/accusative/dative/genitive) was lost, with the first three merging to a bare noun and the last taking the sole form /-s/. Similar things happened with the verbs and syntax, but that would take a long time to get to and isn't the question you asked. Just wanted to clear those two up.

You correctly point out that both French and Germanic (other than English, obviously) have noun gender as a rule. This helps to show us, I think - and this is a somewhat contentious point, but one I and several scholars think is supported by the evidence - that it wasn't the influence of Norman French that caused noun gender to disappear from Middle English, but the change was part of the natural development in English itself.

One quirky feature of Standard German is that you're supposed to use the same gender in a pronoun as the noun it refers to - so, for example, the translation of 'the girl, I saw her' would be Das Mädchen, ich habe -es- gesehen - literally 'it'. That's because the antecedent, 'Mädchen', is a neuter noun, contrary to its natural gender (if you're wondering, this is because the -chen suffix is always neuter). Now, that's the standard formal way of speaking, but in practice you'll find a lot of Germans saying das Mädchen... sie, 'she', especially when the pronoun is a long way away from the antecedent / in a different sentence.

In Old English, however, this tendency was already being lost, even in the same sentence, even when the nouns were near to each other. Thus in an Old English bible translation Jesus says þisne hlaf, -hit- is min lichama, 'this bread, -it- is my body' - even though hlaf, 'bread', is grammatically masculine. This shows that even before the Norman Conquest English was losing some sense of what grammatical gender applied to exactly, and was moving toward natural gender.

Why did this happen in English, say, and not in German? Well, it's very difficult to know with linguistics 'why' something happens - there are rules that show that something's gonna be more likely (for example, Old English speakers wouldn't have made hlaf feminine by this process), but in reality the changes Old English experienced in the generational transitions to Middle English were pretty much idiosyncratic, and difficult to reduce to historical terms (although note that the loss of gender isn't unprecedented in Indo-European - Persian did the exact same thing, Danish, Swedish and Norwegian merged the masculine and feminine, and Romance languages lost the neuter). I'd be wary, as such, of the idea that something 'caused' English to lose its gender, since the convergence of causes is very complicated and occurs in the minds of the speakers rather than as an objective historical fact.

TL;DR neither French nor Germanic languages caused English to lose its system of gender - we already see hints of it in Old English, and it was part of a general grammatical levelling in Middle English which can't be attributed directly to Norman French despite the timeframe. This is a pretty good free source if you're interested in learning more.

If I wasn't clear about anything, please ask, I do think I rambled quite a bit there.

To clear up some initial assumptions in your post:

Since English is heavily influenced by German (although in a very different form than today's German)

English isn't influenced by German. English is a Germanic language, which means it is related to German.

and French

This is for the most part true, due to the Norman Conquest (though this isn't the sole reason for loss of gender). I just want to be sure that we aren't confusing influence with English being a creole of some sorts. French is in a separate branch of the Indo-European family from that of German and English.

Okay, so Old English had gender much the same as you expect with other Indo-European languages. In Old English there were three genders: masculine, feminine, and neuter; four cases: nominative, accusative, genitive, and dative; two numbers: singular and plural. These weren't necessarily determined by biological gender (see hlaefdige fem. 'lady', wif neut. 'woman', wifmann masc. 'woman'). By the tenth century we begin to see a decline in the use of gender, and from the eleventh century until the 14th century all of the dialects of English lose their grammatical gender. The trend was that feminine and neuter animate nouns become masculine, and masculine and feminine inanimate nouns become neuter. Eventually, they disappear altogether with the exception of pronouns and some words (widow/widower) though these are largely based on natural gender rather than grammatical gender. The Norman Conquest facilitated in the shifting of some words gender, though the process started before its happening.

The "how" in this instance is simply language change in action, with genders shifting to neuter and inflectional endings disappearing. The "why" part isn't so clear. There isn't exactly a single factor we can point to and be like "alright, this is why". One big factor is a phonological development that made all unstressed final vowels to /ə/, meaning that the gender distinctions in -e, -o, -a, and -u were lost. However, German underwent a similar change but has retained gender, so it can't be said that the change in English inevitably brought up loss of inflection. Old French influence was there, with their only distinction being between singular and plural. Some say that Scandinavian influence could have brought it about, though it doesn't quite make sense since their influence was localized and their system wasn't too dissimilar. Really, the OE system of gender inflection was very "unstable" (in my opinion) with a lot of overlap between endings. Plus, a lot of nouns in singular gave no hint at what their gender was (gast fem. 'spirit' vs. dust neut.). In OE writings you find a lot of confusion (i.e. speakers had different judgements) over the gender of nouns.

Here's some further reading:

• A Biography of the English Language by C.M. Millward

• http://www.wordorigins.org/index.php/site/comments/loss_of_gender_in_english/

• http://en.wikipedia.org/wiki/Gender_in_English

• Historical Linguistics by Lyle Campbell, this book contains quite a few excerpts and exercises on the loss of gender in English