The new element should come hidden to the slideDown being able to show it, add display:none
online or in the class of the new element:
$(document).ready(function(){
//when the Add Filed button is clicked
$(".adicionarCampo").click(function () {
//Append a new row of code to the "#items" div
$(".lista").append('<div class="nomes" style="display:none"><input class="input-adicional" id="nome" name="nome[]" placeholder="Nome completo" type="text" /><a href="javascript:void(0)" class="delete"></a></div>');
$(".nomes").slideDown("slow");
$("body").animate({scrollTop: $("body").prop("scrollHeight")}, 500);
});
$("body").on("click", ".delete", function () {
$(this).parent("div").slideUp("fast", function() {
$(this).remove()
})
});
});