The Catholicism and Orthodox churches do not recognize the Eucharist of protestant churches. Why not?
As far as both Catholics and Orthodox are concerned, you need a validly ordained priest to confect the Eucharist.(see e.g. CCC, ¶1411) As far as they are concerned, none of the churches that originate from the Reformation period have what is required for a valid priesthood (i.e. bishops in unbroken tactile succession from the Apostles(see CCC, ¶1576)), so the Eucharist cannot be validly celebrated by these churches.
This may look uncharitable on the surface. In reality, the Catholics and Orthodox could not accept the validity of Protestant Eucharists without significantly altering their own doctrine.
On another related point, it's worth noting that Catholics and Orthodox do not even properly consider the Protestant churches "Churches" in the proper sense: they are referred to technically as "ecclesial communities" precisely because they have not preserved the sacramental priesthood and therefore the Eucharist:
According to Catholic doctrine, these Communities do not enjoy apostolic succession in the sacrament of Orders, and are, therefore, deprived of a constitutive element of the Church. These ecclesial Communities which, specifically because of the absence of the sacramental priesthood, have not preserved the genuine and integral substance of the Eucharistic Mystery cannot, according to Catholic doctrine, be called “Churches” in the proper sense. Responses to some questions regarding certain aspects of the doctrine of the Church)
It's clearly because Catholics believe that the bread and wine are the physical Body and Blood of Christ. Protestants do not. So, therefore, it's a fundamentally different ritual in Protestant churches.
The 1896 Papal Bull declaring all Anglican denominations to be "null and void" was after hundreds and hundreds of years of disagreement on the fundamental nature of the eucharist. I am of the opinion, as an Anglican, that it is merely symbolic and demonstrates our devotion to Christ to do exactly as he said to do when we remember him, for thousands of years.
That's why. Personally, I think it's best not to raise the issue with Catholics. It's not a salvation-killer (is there such a thing, ah a topic for another thread) or anything, after all.
The question of validity turns on the view of the Eucharist at Mass. For Catholics, they understand the Eucharist to be an actual sacrifice. And to have an actual sacrifice, one needs a sacrifice at an altar with a duly ordained priest. So yes, "valid" requires a priest, but the whole point is their view that the wine/bread changing into blood/body is to sacrifice it.
As the Second Vatican Council says: "That more perfect form of participation in the Mass whereby the faithful, after the priest's communion, receive the Lord's Body from the same sacrifice, is warmly recommended." CCC 1388
For Protestants, they do not view the wine/bread at communion as a sacrifice. They believe Christ offered Himself once for our sins.
[The high priest Jesus Chrst] [w]ho needeth not daily, as those high priests, to offer up sacrifice, first for his own sins, and then for the people's: for this he did once, when he offered up himself. Heb 7:27
Nor yet that he [Christ Jesus] should offer himself often, as the high priest entereth into the holy place every year with blood of others; For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself. Heb 9:25-26
Note that Christ offered Himself, as against those priests who offer blood of others. Christ no longer suffers.
So, while the answers mention the idea of a so-called "bloodline" of duly consecrated bishops and priests traceable back to apostles, the point of that "bloodline" was not mentioned. The Catholic Church believes their Mass is a sacrifice that requires a valid priest, while Protestants believe in Christ's sacrifice done once and offered solely by Himself.
In 1896, Pope Leo XIII issued a papal bull Apostolicae Curae declaring all Anglican ordinations to be "absolutely null and utterly void". This is because of they changed the form of ordination rites in such a way as to render them invalid. This means the Anglicans don't have valid orders and therefore they cannot confect the Eucharist. Changes to the rites of ordination did not occur during the Great Western Schism.
There are two related aspects
Lack of valid priesthood (also indicated in @lonesomeday's answer). "Only validly ordained priests can preside at the Eucharist and consecrate the bread and the wine so that they become the Body and Blood of the Lord." Catechism of the Catholic Church #1411. The Fourth Lateran Council in 1215 declared "Indeed no one can perform this sacrament except the priest duly ordained" (DH 802 cited in Ludwig Ott's Fundamentals of Catholic Dogma, p 422)
Difference in concept (As outlined in @Ztucker's answer) The question uses the term Eucharist, but that is not broadly used by Protestant Churches. Catholics (and Orthodox) believe the Eucharist to be the Body and Blood of Christ really and substantially present. "That in this sacrament are the true Body of Christ and his true Blood is something that 'cannot be apprehended by the senses,' says St. Thomas, quoted in Catechism of the Catholic Church #1381. In contrast, Protestants either believe it to be purely symbolic or possibly a consubstantial presence, but certainly not a sacrifice.
Both of these concepts must be understood for a complete answer. In contrast to the claim of "ex opere operato" (from the work having been worked), the lack of priesthood alone is not sufficient. If a validly ordained Catholic priest went and celebrated a Protestant communion service (e.g. a Lutheran one), no valid Eucharist would result due to a defect in the required form, matter and intention. For a valid Eucharist to occur, the correct matter (unleavened wheat bread and grape wine without additives) must be used, the correct form (specifics words used in consecration) and the correct intention (at least to do what the Catholic Church intends, but more specifically to turn the bread and wine into the Body and Blood of Christ and offer it). The principle of "ex opere operato" means that the spiritual result of the act depend not on the individual holiness of the minister or recipient but depend on the act itself (the opposite is "ex opere operantis" -- from the work of the one doing the work").