Firstly understand the bond that children have with animals.
Most likely this will be the first relationship where they undoubtedly gave and received unconditional love, outside of parents, sibling etc. This will be the friend who was there for them when they had a bad day at school, when they got sent to their room for bad behaviour and also there for the fun, happy times. So bear in mind this will be a massive wrench for them. Don’t underestimate the love your child would have received and felt. There will also be a level of commitment and responsibility that they will have developed, from grooming and feeding to walking and cuddle time. This will have created a huge bond and time slots in their daily life that will now be gone.
Secondly, when it comes to explaining pet death, stick to the truth. Don’t fluff it up or skirt around the facts. If your pet has an illness and is likely to be put down you will need to start preparing your child. A good way to know what they’re capable of understanding is to let their questions guide you. After the obvious statement from you telling them about what’s going to happen, they will likely have their own questions and that will be enough for them at the beginning. Don’t flood them with information let it gradually come out as the process goes on. If you do decide to euthanize your pet then you must tell your child. Let them know when it will happen and why. They will need to have time to say goodbye. They may not understand that your pets body is unable to carry on and explaining to them that this is the best option for the animal as they will no longer be in pain is a good start.
Adults struggle with the concept of death never mind trying to explain it to a child. No one can say what is best for your child you know them best. However it’s a good idea not to make it too complicated. Maybe start by asking them what their idea of death is. Children have wild imaginations so they may have formed a magical place in their mind that their friend has gone to. Encourage them to describe this place and what their pet is doing. Perhaps you could get them to draw it and keep it on their wall. Honesty is never a bad thing and regardless of your religious or spiritual beliefs you can always say that you’re not 100% sure, if of course you’re not.
In my opinion it’s not a good idea to use the phrase “they’ve gone to sleep” or are being “put to sleep” as this could cause fear for them at night when trying to sleep. Saying “they’ve left us and are in a better place” can make your child angry and feel that their pet had somewhere they wanted to go without them. Keep it simple explain that death can be from an accident or illness and old age. Let them understand that just like humans, an animals body can wear out and weaken from sickness. Talk to them about grief and how it’s ok to be sad and miss their friend at times.
Lastly, don’t try and replace their pet straight away with another one, give it some time. They may feel disloyal if they like the new pet straight after the death of the other one. Talk to your child about the fun things they used to do together and encourage them to speak to you about their favourite memories. Let your child pick their favourite photo of them with their pet and have it framed and put in their bedroom. If they want to hold onto a toy or a collar, don’t disapprove it will help them to feel that they still have some physical contact. Make sure you don’t clear the house of all the pet items straight away, your child will need to feel that not everything has been taken away and that there’s some normal feeling of the pet still around the house. A simple memorial in your garden or at the park is a constructive way to let your child say goodbye. Planting a shrub that they get to pick out for their pet and burying a special drawing they’ve done with it is also a good way for them to have some closure.