As far as I know, some Esperantist were unhappy with some grammar constructs and branched the language.
How did Ido influence Esperanto from the grammatical point of view since then?
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Well, maybe not exactly grammar...
Wikipedia says "Modern Esperanto has received some influence from Ido in areas such as a clarification of the rules for word derivation and suffixes like -oz- ("abundant in") and -end- ("required to")." "In the same manner in which dialects often serve as sources for new words through the literature of ethnic languages, so Ido has contributed many neologisms to Esperanto (especially in poetic substitutes for long words using the mal- prefix)."
Otherwise... The criticism that inspired Ido inspired Zamenhof's proposal of Reformed Esperanto, which ultimately failed.
There seems to be so much more literature in Esperanto than in Ido, so I guess the influence would have had to come from literature. Just from an overview of the Ido I have seen so far, there doesn't seem to be that many differences in grammar itself, so it is unclear what could have influenced Esperanto.
From the "why Ido?"-page itself: "Ido is also not encumbered by the peculiar idiomatic uses of the 'accusative' ending which occur in Esperanto (for example: la duan fojon, tagon post tago, unu matenon, iri Parizon) and which add unnecessary complication to that language. In these cases the ending -n does not indicate an accusative! In Ido, a preposition is used where appropriate (for example, 'ye la lasta foyo')."
The icxismo proposal is of course very similar to the Ido idea of having "-ulo" and "-ino" as male and female endings respectively. But I wouldn't say that that is an "Ido" idea.
If we compare the Fundamental grammar paragraph after paragraph,
§1: Ido added a kind of plural article le - not in Esperanto.
§2: Ido has plural -i, the accusative is restricted to seldom cases. Both didn't make it into E-o.
§3: Ido has no adjective agreement, E-o still has.
§4: Ido added an ordinal suffix -esm-, which didn't make it into E-o.
§5: Ido has a sexsymmetrical system, while esperantists are still fighting how to express a single human neutrally. Ido's lu is not an option, however (for lu/i 'to rent').
§6: Ido introduces a synthecial passive -es- and a pluperfect -ab-. One could argue that the expanding passive use of E-o -iĝ- resembles Ido -es-, but this would be speculative; the evolution can stem from the linguistic system of E-o itself.
§7,8: Same in Ido
§9: Ido introduced some irregularities, which didn't make it into E-o. Same thing with the abolition of the diacritics.
§10: Again some exceptions in Ido, which didn't make it into E-o.
§11: Superficially same in both languages, but in fact Ido completely changed the word formation system (making affixoids bound morphemes and more of them obligatory; altering the lexical classes). A few elements made it into E-o (as mentioned in the other responses), but not the system as a whole.
§12: Same in Ido.
§13, 14: Abolished in Ido, still fine in Esperanto.
§15: I can't really tell whether Ido has any rule on the borrowing new roots.
§16: Same in Ido, with orthographical complications. Additionally the adjectival -a may be ellipsed, which didn't make it into E-o.
the influence of Ido on the core grammar of Esperanto is nil.
Also not exactly grammar, but some of the anti-mal- words introduced as "poetic" words in the 1920s are the same in Ido (olda, kurta, basa, streta but frida vs kolda, liva vs sinistra), which had foregrounded the criticism of such words; and some of their proponents, like Kalocsay, had spent time in Ido.